In this article we will discuss about:- 1. Meaning of Condensation 2. Laminar Film Condensation on a Vertical Plate 3. Turbulent Film Condensation.
Meaning of Condensation:
Fluid in a gaseous or vapour phase changes to a liquid state with the liberation of heat from the vapour.
When a vapour is in contact with a surface whose temperature ts is lower than the saturation temperature tsat corresponding to the vapour pressure, the condensation sets in and the vapour changes to liquid phase. The condensation of vapour liberates latent heat and there is heat flow to the surface. The liquid condensate may get somewhat sub-cooled by contact with the cooled surface and that may eventually result in more vapour to condense on the exposed surface or upon the previously formed condensate.
Depending upon the behaviour of condensate upon the cooled surface, the condensation process has been categorised into the following distinct modes:
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(i) Film Condensation:
The liquid condensate wets the solid surface, spreads out and forms a continuous film over the entire surface. The liquid flows down the cooling surface under the action of gravity and the layer continuously grows in thickness because of newly condensing vapours. The continuous film offers thermal resistance and restricts further transfer of heat between the vapour and the surface.
Film condensation usually occurs when a vapour, relatively free from impurities, is allowed to condense on a clean surface.
(ii) Drop-Wise Condensation:
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The liquid condensate collects in droplets and does not wet the solid cooling surface. The droplets develop in cracks and pits on the surface, grow in size, break away from the surface, knock off other droplets and eventually run off the surface without forming a film.
A part of the condensation surface is directly exposed to the vapour without an insulating film of condensate liquid. Evidently there is no film barrier to heat flow and higher heat transfer rates are experienced; heat transfer fluxes of the order of 750 kW/m2 have been obtained with drop-wise condensation.
Dropwise condensation has been observed to occur either on highly polished surfaces, or on surfaces contaminated with impurities like fatty acids and organic compounds. Dropwise condensation gives coefficient of heat transfer generally five to ten times larger than with film condensation.
Because of potential performance gain, dropwise condensation is provoked artificially by surface coatings, called promoters that inhibit wetting. Silicons, teflons and an assortment of waxes and fatty acids are often used for this purpose. These substances are either applied to the heat transfer surface or introduced into the vapour.
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However, the phenomenon is highly unstable as these coatings gradually lose their effectiveness due to oxidation, fouling or outright removal and the surfaces become wetted when exposed to condensing vapour over an extended length of time. Consequently film condensation is generally encountered in industrial applications and is usually planned for condenser design calculations.
Laminar Film Condensation on a Vertical Plate:
Consider the process of film condensation occurring on the surface of a flat vertical plate as depicted in Fig. 12.2. The coordinate axes of the system have been so chosen that the origin ‘o’ is at the upper end of the plate, the x-axis lies along the vertical surface (positive direction of x measured downward) and the y-axis is perpendicular to it.
The thickness of liquid film, which is zero at the upper end of plate gradually increases as further condensation occurs at the liquid-vapour interface and attains its maximum value at the lower end of the plate. The vertical plate has height l, width b, and 8 denotes the thickness of film at a distance x from the origin.
An estimate of the heat transfer coefficient for the liquid film can be made by setting up expressions for the velocity distribution, the mass flow rate and heat flux through the layer.
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The analysis is based on the findings of Nusselt (1916) and it makes the following assumptions:
i. The liquid film is in good thermal contact with the cooling surface and therefore temperature at the inside of the film is taken equal to the surface temperature ts. Further, the temperature at the outer surface of the film (interface of liquid and vapour) equals the saturation temperature tsat at the prevailing pressure.
ii. The condensate film is so thin that a linear temperature variation exists between the plate surface and the vapour conditions.
iii. The physical parameters (the thermal conductivity k, the dynamic viscosity µ and the density ρ) of the condensate film are independent of temperature.
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iv. Vapour density is small compared to that of the condensate.
v. The vapour delivers only the latent that (i.e., there is no under cooling of condensate), and the heat flow across the plate is by conduction.
vi. An element of fluid mass within the film is influenced only by the viscous shear and the gravitational forces; the forces of inertia appearing in the condensate film are disregarded.
vii. There is no velocity gradient at the liquid-vapour interface and obviously the viscous shear at the phase interface is negligible.
viii. The condensing vapour is entirely clean and free from gases, air and non-condensing impurities.
ix. Drainage of condensate film along the vertical surface is by the action of gravity and is through a laminar motion.
x. Radiation between the vapour and the liquid film; horizontal component of velocity at any point in the liquid film; and curvature of the film are considered negligibly small.
Velocity Distribution:
An equation for the velocity distribution u as a function of distance y from the wall surface can be setup by considering the equilibrium between the gravity and viscous forces on an elementary volume (b dx dy) of the liquid film.
Gravitational force on the element = ρg (b dx dy)
Viscous shearing stress on the element face at y = µ du/dy
Mass Flow Rate:
The downward flow of the liquid at any elevation x (i.e. over the layer of thickness δ) is mass flow rate = mean flow velocity × flow area × density-
The mass flow is thus a function of x; this is so because the film thickness δ is essentially dependent upon x.
An increase in the mass flow rate of condensation during downward flow of condensate from x to x + dx can be worked out by differentiating equation 12.4 with respect to x or δ.
Heat Flux:
The heat flow rate into the film, dQ, equals the rate of energy release due to condensation at the surface. Thus,
Where, hfg is the latent heat of condensation.
Nusselt presumed that the heat released during condensation flows only by conduction through the film.
Evidently the film thickness increases as the fourth root of the distance down the surface; the increase is rather rapid at the upper end of the vertical surface and slow thereafter.
Film Heat Transfer Coefficient:
Nusselt had presumed that heat flow from the vapour to the surface is by conduction through the liquid film, i.e.,
Thus at a definite point on the heat transfer surface, the film coefficient hx is directly proportional to thermal conductivity k and inversely proportional to thickness of film δ at that point.
Substituting the value of film thickness δ from equation 12.8,
Local heat transfer coefficient at the lower edge at plate, i.e., at x = I
Undoubtedly the rate of condensation heat transfer is higher at the upper end of the plate than at the lower end.
By integrating the local value of conductance (equation 12.10) over the entire length I of the plate, we get the average heat transfer coefficient;
Where, h is the local heat transfer coefficient at the lower edge of the plate.
Then it follows from equation 12.9 that –
Where, δ(l) is the film thickness at the lower end of the plate. Obviously the average heat transfer coefficient is 4/3 times the local heat transfer coefficient at the trailing edge of the plate.
Equation 12.12. is usually written in the form,
The Nusselt solution derived above is an approximate one because of the assumptions admitted in the statement of the problem. Experimental results have shown that the Nusselt equation is conservative; it yields results which are approximately 20% lower than the measured values. Accordingly, use of a value of 1.13 in place of the coefficient 0.943 has been recommended by McAdams.
The results concerning development of film condensation along a vertical flat plate indicate that the thickness of film increases with increase in plate height, (equation 12.8). Since the thermal resistance increases with film thickness, a decrease in heat transfer coefficient is expected and that is evident from the relations 12.10 and 12.15.
The dependence of these parameters on height of plate has been shown graphically in Fig. 12.3. Further the convection coefficient increases with temperature difference (tsat – ts). This may be attributed to an increase in the film thickness which results from increase in condensate rate at high temperature differences.
Whilst computing the average coefficient vide equation 12.15, all the fluid properties are evaluated at the film temperature tf = (tsat + ts)/2, and the vapour latent heat of condensation is evaluated at tsat. Having thus determined the average condensation coefficient, the following expressions are used to obtain the total heat transfer and the total condensation rate.
For inclined flat surfaces, the gravitational acceleration in the basic Nusselt equation is replaced by a projection of the gravity acceleration vector on the y-axis: gy = g sin ϴ where ϴ is the inclination angle with the horizontal. This yields the following expression for film condensation on a flat inclined surface;
Equation 12.19 must be used with caution for small values of inclination ϴ; its accuracy becomes poor as the angle of inclination approaches the horizontal.
Turbulent Film Condensation:
The character of condensate film can range from laminar to highly turbulent. The liquid flows in laminar film at the upper end of the plate, then becomes undulating in the middle section and finally flows in a turbulent state. When turbulence sets in, the condensate film no longer offers as high a thermal resistance as it does in laminar film. Heat is then transferred not only by conduction but also by eddy diffusion which is a characteristic of turbulence.
Obviously the turbulent condensate film results in increased convective coefficients. Turbulent films are usually encountered when the film thickness becomes appreciable, i.e., when the condensation rates are large or when the condensation surface has substantial length.
The parameter indicating the commencement of turbulent flow is the Reynolds number conventionally defined as Re = Vdeq ρ/µ where the characteristic length is the equivalent diameter deq = 4A/P. For a vertical surface the flow area A is b δ and the wetted perimeter P of the solid interface is simply the width b of the plate.
The transition from laminar to turbulent flow occurs at a critical Reynolds number of 1800. For turbulent film condensation on vertical surfaces, Kirkbride has suggested the following correlation for the average heat transfer coefficient.
The distinguishing features between the laminar and turbulent film condensation as conveyed by equations 12.14 and 12.21 are-
(i) In the laminar film, the average film coefficient decreases with distance I. This is due to gradual increase in the thickness of laminar film.
(ii) In the turbulent region, the average film coefficient increases with distance I. This is due to eddies which promote convection. Heat is then transferred not only by conduction but also by eddy diffusion.