Investigation of Heat Transfer in Dropwise Condensation Based on Similarity Theory
Abstract Generated abstract
This paper develops a similarity-theory treatment of heat transfer during steady dropwise condensation of pure vapor on cooled nonwetting surfaces. Starting from governing relations for heat transfer through drops, liquid heat conduction and motion, continuity, and phase-boundary conditions, the authors derive a criterial equation involving a modified Nusselt number and dimensionless groups including Galileo, Peclet, Prandtl, latent-heat, and pressure-related criteria. Photographic analysis is used to estimate the fraction of surface covered by droplets, and experiments on water vapor at atmospheric pressure with several hydrophobizing coatings provide 69 data points. For these conditions, the data are summarized by a simplified relation between the modified Nusselt criterion and the product of the Peclet and latent-heat criteria, which is proposed for calculating heat transfer in dropwise condensation of pure steam at 1 atmosphere.
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STUDY OF HEAT TRANSFER DURING DROPWISE CONDENSATION BASED ON SIMILARITY THEORY
G. I. Fuks, V. V. Ivanov
The article presents the results of theoretical and experimental studies of heat transfer during dropwise condensation of vapor. A system of similarity criteria has been obtained for generalizing experimental data.
Dropwise condensation, occurring on cooled nonwetting surfaces, is accompanied by very high heat-transfer coefficients (up to 20,000 kcal/m²·h·°C). This is why this form of condensation is attracting ever increasing attention from researchers [1, 2, 3].
Interesting attempts at a theoretical study of heat exchange during dropwise condensation were made by Fatica and Katz [4] and by Sugawara and Michiyoshi [5]. They consider a vertical strip on the cooling surface corresponding to the trace of a drop that has just rolled down. In doing so, the following assumptions are made: all drops on this strip begin to grow simultaneously and simultaneously reach the same critical diameter; at any instant during growth the drops have the same size and the shape of a spherical segment; the vapor is motionless; and thermal resistances at the vapor–drop surface boundary and between the vapor and the “bare” sections of the wall are absent. Then the average heat-transfer coefficient is determined by the expression [5]
\[ \alpha = \frac{V_c}{ \displaystyle \int_0^{V_c} \frac{A}{\psi K_d + K_0(1-A)}\, dV }, \tag{1} \]
where \(V\) is the variable volume of condensate on the cooling surface during drop growth; \(V_c\) is the volume of condensate at the end of the drop growth period; \(A\) is the ratio of the area covered by drops to the entire area of the cooled surface; \(K_d\) is the heat-transfer coefficient through a growing drop; \(K_0\) is the heat-transfer coefficient through the wall to the cooling water; \(\psi\) is a correction coefficient.
However, equation (1) cannot be used in practical calculations, since, first, it contains the undetermined constants \(V_c\) and \(\psi\), and second, the functional relationship of \(K_d\) and \(A\) to \(V\) is unknown.
In the present work an attempt is made to calculate heat transfer under conditions of steady dropwise condensation of pure vapor on the basis of similarity theory.
The basic system of equations describing heat exchange in dropwise condensation of pure vapor has the following form.
I. Equation for the heat-transfer coefficient (1);
II. Equation for the coefficient of heat transfer through a drop
\[ K_d(t_n-t_k)f=-\lambda\left(\frac{\partial t}{\partial n}\right)f, \]
where \((t_n-t_k)\) is the temperature difference between the surface of the drop and its base, \(\lambda\) is the thermal-conductivity coefficient of the condensate, \(\left(\dfrac{\partial t}{\partial n}\right)\) is the temperature gradient in the liquid phase at the boundary with the wall, \(f\) is the area of the base of the drop;
III. Equations of heat conduction, motion, and continuity of the liquid phase
\[ a\nabla^2 t=\frac{Dt}{d\tau}, \]
\[ \vec{g}\rho-\operatorname{grad}p+\mu\nabla^2\vec{W}=\rho\frac{D\vec{W}}{d\tau}, \]
\[ \operatorname{div}\vec{W}=0; \]
IV. Conditions of phase interaction at the condensate–vapor boundary
\[ t_n=t_{gr}, \]
\[ p_{gr}=p''+2\frac{\sigma}{R}, \tag{IV-a} \]
\[ r\rho W_k=-\lambda\left(\frac{\partial t}{\partial n}\right)_{gr}, \tag{IV-b} \]
where \(\sigma\) is the surface tension of the condensate, and \(W_k\) is the growth rate of a drop of radius \(R\).
The conditions of uniqueness of this problem include the physical constants, the difference between the saturation temperature and the temperature of the cooled wall: \(\Delta t=t_n-t_{cm}\) (or the heat-flux density \(q=\alpha\Delta t\)), and the geometric dimensions of the system. Since a process that has become steady (with respect to large time intervals) is considered, the initial conditions are not specified.
Processing equations I–IV by the method of similarity theory, we obtain the following criteria.
From equations I–II we find the modified Nusselt criterion
\[ Nu_k=\frac{\alpha-\dfrac{1-A}{A}K_0}{\lambda}\,l \]
From equations III—the homochronicity, Froude, Euler, Reynolds, Fourier, and Peclet criteria. From equation IV-a, a criterion characterizing-
the ratio of the saturation pressure to the pressure jump at the phase boundary
\[ K_p=\frac{p l}{\sigma}. \]
From equation IV-6, the criterion is
\[ \frac{r\gamma W_\kappa l}{\lambda \Delta t} = \frac{W_\kappa l}{a}\cdot \frac{r}{C_p \Delta t} = Pe \cdot K \]
In studying quantities averaged over time, the homochronicity and Fourier criteria, which contain time, are discarded. The Euler criterion also drops out \(\left(Eu=\frac{\Delta p}{\rho W^2}\right)\), since the quantity \(\Delta p\) does not enter into the conditions of uniqueness of the given problem.
Instead of the Froude criterion, let us introduce the Galileo criterion, identical with it,
\[ Ga=Fr\cdot Re^2=\frac{g l^3}{\nu^2}. \]
Bearing in mind that \(Pe\), \(Pr\), and \(Re\) are related by the relation \(Pe=Re\cdot Pr\), we omit \(Re\), introducing the criterion \(Pr\).
Thus, the criterial equation for heat transfer during dropwise condensation of pure vapor will have the form
\[ Nu_\kappa= \frac{\frac{1-A}{A}K_0}{\lambda}\,l = \Phi(Ga,\ Pe,\ Pr,\ K,\ K_p) \tag{2} \]
To determine the relative area covered by drops, a quantitative processing was carried out on a large number of photographs of the process of dropwise condensation (some of the photographs are given in Fig. 1). As it turned out, the value \(A\) lies within the limits from 0.35 to 0.55. Similar values of \(A\) were obtained in the experiments of Hampson [3], Fatica and Katz [4], and Sugawara and Michiyoshi [5].
For simplicity of calculations it may be assumed that the area occupied by drops amounts on average to 45% of the entire cooling surface. Then the Nusselt criterion in equation (2) will take the form:
\[ Nu_\kappa=\frac{\alpha-1.22K_0}{\lambda}\,l \]
Visual observations and photographs of the process of dropwise condensation show that the centers of condensation are distributed randomly over the surface and, consequently, uniformly. Under these conditions, the intensity of heat transfer should not depend on the linear dimension of the surface. Here it is expedient to take as the characteristic dimension the critical droplet diameter \(D_\kappa\), which can be determined by the relation [6]
\[ D_\kappa=0.018\beta\sqrt{\frac{\sigma}{\gamma'-\gamma''}}, \]
where \(\beta\) is the contact angle, \(\sigma\) is the surface tension of the condensate, and \(\gamma'-\gamma''\) is the difference between the specific weights of the condensate and the vapor. Experiments carried out with kerosene, SU machine oil, oleic and stearic acids dissolved in ether showed that, under conditions of stable dropwise condensation, the angle \(\beta\) varies within the limits
approximately from 90 to 100° and is practically independent of the hydrophobizer. Therefore, in equation (2), instead of \(D_k\) we introduce the determining dimension
\[ l=\sqrt{\frac{\sigma}{\gamma' - \gamma''}}, \]
which is proportional to \(D_k\), but does not contain \(\beta\).
Fig. 1. Photographs of the process of dropwise condensation on a cooled flat wall.
As for the variable \(W\), which enters the Peclet criterion, it is assumed that this empirical quantity, having the dimension of velocity, is equal to \(\dfrac{D_k}{\tau_c}\), where \(\tau_c\) is the lifetime of a droplet on the cooling surface. The use of such a quantity is due to the following. The process of dropwise condensation is in many respects analogous to the process of nucleate boiling, for which the value \(\dfrac{D_0}{\tau_c}\), where \(D_0\) is the detachment diameter of the vapor bubble, does not depend on the heat load of the heating surface or on the hydrodynamic conditions of the process [7]. Gelman’s experiments [6] on dropwise condensation of mercury vapor showed that, with an increase in heat flux, the frequency of droplet formation \(1/\tau_c\) is practically constant, although some increase in \(D_k\) was noted. In our experiments on dropwise condensation of water vapor, carried out over the same range of variation of the heat load, no increase in \(D_k\) was observed.
It may be assumed that, for dropwise condensation as well, the quantity
\[ \frac{D_k}{\tau_c} \]
does not depend on the heat load or on the hydrodynamic conditions of the process, although, apparently, as in the case of nucleate boiling, it varies with pressure. This may serve as a basis for including
\[ \frac{D_k}{\tau_c}, \]
as a peculiar physical constant, in the conditions of uniqueness and in the determining Peclet criterion.
From equation (2), as a particular case, the criterion equation for heat transfer in film condensation of pure vapor is obtained:
\[ \frac{\alpha l}{\lambda}=\Phi_1\left(\frac{g l^3}{\nu^2},\ \frac{\nu}{a},\ \frac{r}{C_p \Delta t}\right). \]
Indeed, in film condensation \(A=1\); therefore the modified Nusselt criterion becomes the ordinary one, \(Nu=\dfrac{\alpha l}{\lambda}\). The criteria \(Pe\) and \(K_p\) entering equation (2) drop out, since the first includes the quantity \(W\), which is not included in the conditions of uniqueness of the film-condensation process, while the second is obtained from relation IV—a, valid only for spherical drops. Since the critical diameter of a drop plays no role in the process of film condensation, here the height (for vertical tubes) or diameter (for horizontal tubes) is taken as the characteristic dimension.
On the basis of equation (2), experimental data on heat transfer in dropwise condensation of pure vapor should be processed.
Analysis of the experimental data obtained on dropwise condensation of water vapor at a pressure \(p=1\) atm made it possible to obtain the following relationship
\[ Nu_k=C\,Pe\cdot K \tag{3} \]
Since the experiments were carried out at one pressure and with one liquid, the influence of the criteria \(Ga\), \(Pr\), \(K_p\) on the process under consideration could not be established. Since in the experiments the value
\[ \frac{D_k}{\tau_c} \]
was not determined separately, comparison of the experimental data obtained at atmospheric pressure made it possible to determine only the product \(C\dfrac{D_k}{\tau_c}\). This quantity is numerically equal to \(0.5\ \text{m/h}\).
Experimental apparatus and conduct of the experiments
The study of heat transfer in dropwise condensation was carried out on an apparatus whose diagram is shown in Fig. 2. The body 1 of the condenser was made from a tube 182/174 mm in diameter and 262 mm high. Inside the body, forming a steam jacket, cylinder 2, 144/140 mm in diameter, was inserted. Dropwise condensation occurred on brass tube 3 of square cross section \(27\times27\) mm with a wall thickness of 1 mm. In this case the working surface was \(0.0225\ \text{m}^2\). The flat face of the test tube, facing sight glass 4, was provided with a linear scale (one division visible in the photographs corresponds to 5 mm). This made it possible, by quantitative process-
...of the photographic images of the process, to determine the area occupied by the drops and the sizes of the drops. The cooling system was made according to a “tube-in-tube” scheme. The temperature of the cooling water at the inlet and outlet of the condenser was measured with laboratory thermometers with a scale division of 0.1°C. The water flow rate was determined by the volumetric method. Water vapor, slightly superheated by means of an electric heater, after passing through desuperheater 5, was fed through two tubes into the working volume of the condenser and into the steam jacket. The steam temperature was measured by a copper-constantan thermocouple installed after the desuperheater. The condensate forming on the cooled wall flowed into funnel 6
Fig. 2. Diagram of the experimental apparatus.
and was then discharged through glass tube 7 into a measuring vessel. Tube 7 had two valves at its ends, adjusted so that the vapor, together with the condensate, would not break through to the outside. The condensate flow rate was determined by the weight method. To prevent condensate formed on the cover of the apparatus from entering the experimental tube, a condensate-drain cap 8 was provided. The temperature of the working surface was measured at seven points by means of copper-constantan thermocouples. For each experiment, the mean temperature was determined
wall as an area-weighted average. The thermoelectromotive force of the thermocouples was measured with a PP-type potentiometer; no temperature oscillations associated with the periodic formation and detachment of drops were observed. The mean temperature head between the vapor and the wall varied in the experiments from 1 to 20°C. This corresponded to a change in the cooling-water flow rate from 70 to 830 kg/h and in velocity from 0.1055 to 1.25 m/s. To eliminate the ingress of air into the vapor space, all experiments were carried out at a pressure in the condenser 15–20 cm of water column above atmospheric pressure and with vent 9 open.
As nonwetting coatings providing stable dropwise condensation, kerosene, SU machine oil, oleic acid, and stearic acid dissolved in ether were used. Aniline and ortho-xylol gave unstable dropwise condensation, readily passing over into mixed condensation with a corresponding decrease in heat transfer. The hydrophobizing agents were applied to the cooled surface with a brush after 3–4 experiments. Owing to its small thickness, the thermal resistance of the nonwetting coating had no effect on the heat transfer.
Fig. 3. Comparison of experimental data with equation (3).
○—kerosene; ●—oleic acid;
▽—stearic acid dissolved in ether;
▼—SU machine oil.
The heat-transfer coefficient for dropwise condensation was calculated from the formula
\[ \alpha=\frac{G_k(i_n-i_k)}{F(t_n-t_{cm})}, \]
where \(G_k\) is the hourly condensate flow rate, \((i_n-i_k)\) is the difference between the heat contents of the vapor and the condensate at the inlet to and outlet from the condenser, and \(F\) is the working surface. At the same time, for checking, the heat flux and the heat-transfer coefficient \(\alpha\) were calculated from the flow rate and the increase in temperature of the cooling water.
... by the change in temperature of the cooling water. The difference between the measurement results obtained by these two methods did not exceed 8%. The coefficient \(K_0\) entering equation (2) was found from the equation
\[ K_0 = \frac{G_k\,(i_p - i_k)}{F\,(t_{\mathrm{st}} - t_{\mathrm{v}})}, \]
in which the quantity \(t_{\mathrm{st}} - t_{\mathrm{v}}\) is the mean temperature difference between the wall surface and the cooling water.
Figure 3 presents a summary treatment of all the experimental data in the form of the dependence of \(Nu_k\) on the product \(C Pe K\). The graph obtained shows that the experimental points (69 experiments in all) are distributed quite satisfactorily around the straight line corresponding to equation (3). Therefore, formula (3) may be recommended for calculating heat transfer in dropwise condensation of pure steam \((p = 1\ \text{ata})\).
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