STUDY OF HEAT TRANSFER DURING DROPWISE CONDENSATION OF STEAM

V. V. IVANOV

(Presented by Professor Doctor G. I. Fuks)

Calculation formula

for the heat-transfer coefficient

In deriving the equation for the heat-transfer coefficient in dropwise condensation, $\bar{\alpha}$ [1], it was assumed that the process is steady-state and that the drops themselves are immobile. Such a formulation of the problem made it possible to express $\bar{\alpha}$ in terms of the geometrical constants characterizing the process: the area occupied by drops, $F_k$, the mean drop size $R_{\mathrm{cp}}$, and the wetting angle $\beta$. The formula obtained has the form

\[ \bar{\alpha}=\frac{F}{F_k}\left[\frac{\lambda f(\beta)}{R_{\mathrm{cp}}\sin\beta}+K\right]-K, \tag{1} \]

where $F$ is the condensation surface, $\lambda$ is the thermal-conductivity coefficient of the condensate, $f(\beta)$ is a dimensionless function depending on $\beta$, and $K$ is the coefficient of heat transfer through the wall to the cooling water.

The use of relation (1) in practical applications is hindered by the absence in it of quantities specified in heat-transfer calculations: $(t_n-t_{st})$, the temperature head between the steam and the cooled wall, or the heat-flux density $q_k=\alpha(t_n-t_{st})$. To remove this uncertainty, it is expedient to modify equation (1), constructing it in the form of a function $\bar{\alpha}$ of $(t_n-t_{st})$. For this purpose, let us first consider the growth of drops during condensation.

The heat-balance equation for a growing drop may be written

\[ Q_{pk}=L\gamma\frac{dR}{d\tau}. \]

Here $Q_{pk}$ is the heat flow from the steam to the drop, $L$ is the latent heat of vaporization, $\gamma$ is the specific weight of the condensate, and $\dfrac{dR}{d\tau}$ is the rate of growth of the drop.

We shall express the quantity $Q_{pk}$ by means of the heat-transfer coefficient through a single drop $K_{\mathrm{ed}}$ [1]

\[ Q_{pk}=K_{\mathrm{ed}}(t_n-t_k)=\frac{\lambda f(\theta)}{R\sin\beta}\,(t_n-t_k), \]

where \((t_n-t_k)\) is the temperature difference between the surface and the base of the drop, and \(R\) is the drop radius.

Then

\[ \frac{\lambda f(\theta)}{R\sin\beta}\,(t_n-t_k)=L\gamma\,\frac{dR}{d\tau}. \]

We separate the variables \(R\) and \(\tau\) and integrate the last equation over the limits from \(R=0\) to \(R=R_{\max}\) and from \(\tau=0\) to \(\tau=\tau_c\), where \(\tau_c\) is the lifetime of a drop on the cooling surface:

\[ \frac{\lambda f(\theta)}{\sin\beta}\,(t_n-t_k)=L\gamma\,\frac{R_{\max}^2}{2\tau_c}. \]

Replacing \(R_{\max}\) by the mean size of the drops located on the surface, \(R_{\max}=C_1R_{\mathrm{av}}\), where \(C_1>1\), we shall have

\[ \frac{\lambda f(\theta)}{R_{\mathrm{av}}\sin\beta} = \frac{L\gamma C_1}{2(t_n-t_k)}\cdot\frac{R_{\max}}{\tau_c}. \tag{2} \]

The process of dropwise condensation is in many respects analogous to the process of nucleate boiling, for which the value \(D_0/\tau_c\), where \(D_0\) is the departure diameter of a vapor bubble, does not depend on the heat load of the heating surface [2]. Gelman’s experiments on the dropwise condensation of mercury vapor showed that, as the heat flux increases, the frequency of drop formation \(1/\tau_c\) is practically constant, although some increase in \(D_{\max}\) was noted [3]. In our experiments on the dropwise condensation of water vapor, carried out over the same range of heat-load variation, no increase in \(D_{\max}\) was observed. It may be assumed that, for dropwise condensation as well, the constant

\[ C_2=\frac{R_{\max}}{\tau_c} \]

does not depend on the load, although apparently, as in the case of nucleate boiling, it must vary with pressure.

Combining now (1) and (2), we obtain

\[ \alpha=\frac{F}{F_k}\left[\frac{L\gamma C_1C_2}{2(t_n-t_k)}+K\right]-K. \]

The magnitude of the temperature head between the surface of the drops and their base, \((t_n-t_k)\), is found from the expression for the mean temperature difference between the vapor and the cooled wall [4]:

\[ (t_n-t_{\mathrm{st}})= \frac{(t_n-t_k)F_k+(t_n-t_0)(F-F_k)}{F}, \]

where \(t_0\) is the mean temperature of the surface not occupied by drops. Under conditions of dropwise condensation of pure vapor,

\[ t_n\approx t_0 \quad\text{and}\quad (t_n-t_k)=\frac{F}{F_k}\,(t_n-t_{\mathrm{st}}). \]

Comparison of the experimental data on dropwise condensation at a pressure \(p = 1\) atm made it possible to determine the product \(C_1 C_2\) m/hour. This quantity is numerically equal to unity. For simplicity of calculation it may be assumed that the area occupied by drops averages 45% of the entire cooling surface. Then the calculation formula for the heat-transfer coefficient in dropwise condensation \((p = 1\ \text{atm})\) takes the form

\[ \alpha = \frac{L \cdot \gamma}{2(t_{\mathrm{n}} - t_{\mathrm{st}})} \, 1.22K \tag{3} \]

The quantity \(\gamma\) entering into (3) is taken at the mean temperature of the condensate \(t_m = \frac{1}{2}(t_{\mathrm{n}} + t_{\mathrm{st}})\), and \(L\) at the saturation temperature.

Experimental Apparatus and Conduct of the Experiments

The experimental study of the process of dropwise condensation and the verification of the theoretical relations obtained for the heat-transfer coefficient were carried out on an apparatus whose schematic diagram is given in Fig. 1.

Water vapor from boiler 1, after passing through superheater 2—a section of pipe with an adjustable electrical resistance wound on it—entered steam dryer 3. From there, through two pipes, the steam was supplied to condenser shell 4 and to steam jacket 5. The steam temperature was measured by a copper-constantan thermocouple installed after the steam dryer. This eliminated stratification of the steam flow and ensured good mixing. The experimental condenser consisted of a brass tube of square cross section 6, 27 mm wide, with one face turned toward observation glass 7. The working surface was provided with a linear scale and amounted to \(0.0226\ \text{m}^2\) with a wall thickness of 1 mm. The use of such a tube made it possible, first, to obtain photographs of the dropwise-condensation process on the flat wall and then to calculate the area occupied by the drops and their average size; second, photographs of the drops on the side faces made it possible to determine the wetting angle. The condensate forming on the cooled wall flowed into funnel 8 and was then discharged outward through glass tube 9. The latter had two valves at its ends, adjusted so that steam would not escape outward together with the condensate. In its upper part the experimental tube was provided with condensate-drainage cap 10. This prevented condensate forming on the cover of the apparatus from reaching the cooled wall. The temperature of the condensation surface was measured at seven points by means of copper-constantan thermocouples led out to a potentiometer through special sealed plugs. The condensate flow rate was determined by the weighing method. A U-shaped water manometer 11, connected to the base of drainage tube 12, served to measure the pressure in the apparatus.

Cooling water was supplied from the water mains to upper tank 13, from which, under constant head, it entered the condenser. The cooling system of the experimental tube was made according to the “tube-in-tube” scheme. Such an arrangement made it easy to remove, together with the cover, the experimental tube soldered to it, in order to apply the hydrophobizing agent. The temperature of the cooling water at the inlet and outlet was measured by laboratory thermometers installed in sleeves. The cooling-water flow rate was measured by means of calibrated measuring tank 14. All parts of the apparatus in contact with steam and water were made of copper in order to avoid rusting.

Fig. 1. Schematic diagram of the experimental setup.

The experiments were carried out in several series with different hydrophobizing agents. Each series included a certain number of tests differing from one another in temperature heads \((t_{n} - t_{st})\). The wall temperature was maintained at a prescribed value by appropriate regulation of the cooling water. To eliminate the ingress of air into the steam space, all tests were conducted at a pressure in the condenser \(15\text{--}20\ \mathrm{cm}\) of water column above atmospheric pressure and with the air vent 15 open for purging.

Kerosene, SU machine oil, oleic acid, stearic acid dissolved in ether, aniline, and orthoxylol were used as hydrophobizing agents. The hydrophobizing agent was applied to the cooled wall with a brush, and no less frequently than every 3–4 tests.

The heat-transfer coefficient in dropwise condensation is calculated by the formula

\[ \alpha = \frac{G_{k}(i_{n}-i_{k})}{F(t_{n}-t_{st})}. \tag{4} \]

Here \(G_{k}\) is the condensate flow rate, and \((i_{n}-i_{k})\) is the mean difference between the heat contents of the steam and condensate at the inlet to and outlet from the condenser.

At the same time, for control, \(\alpha\) was determined from the relation

\[ \alpha = \frac{G_{v}(t_{out}-t_{in})}{F(t_{n}-t_{st})}, \tag{5} \]

Fig. 2. Dependence of the heat-transfer coefficient \(\alpha\) \([\mathrm{kcal}/\mathrm{m}^{2}\cdot\mathrm{h}\cdot\mathrm{deg}]\) on \(t_{n}-t_{st}\). \(\bigcirc\)—kerosene; \(\bullet\)—oleic acid; \(\overline{\ }\)—stearic acid dissolved in ether; \(\blacktriangledown\)—SU machine oil. The solid line corresponds to equation (4).

where \(G_{v}\) is the cooling-water flow rate, and \((t_{out}-t_{in})\) is the mean difference between the temperatures of the cooling water at the outlet from and inlet to the condenser.

As the experiments showed, the difference in the values of \(\alpha\) calculated from the equa-

b

a

Fig. 3. Photographs of the process of dropwise condensation: a—oleic acid; b—stearic acid.

…to equations (4) and (5) did not exceed 8%. The coefficient \(K\) entering into (3) was found from the equation

\[ K=\frac{G_k(i_n-i_k)}{F(t_{cm}-t_{\mathrm{в}})} , \]

in which \((t_{cm}-t_{\mathrm{в}})\) is the mean temperature difference between the wall surface facing the steam and the cooling water.

Figure 2 presents the dependence of the heat-transfer coefficient \(\alpha\) on the temperature head \((t_n-t_{cm})\). The hydrophobizing agents used were kerosene, machine oil “CV,” oleic and stearic acids, which gave stable dropwise condensation over the course of 3–4 experiments. As it turned out, the type of hydrophobizing agent does not affect the heat exchange. According to the experimental data, the heat-transfer coefficient under conditions of stable dropwise condensation decreases substantially as the temperature head increases. The obtained experimental dependence of \(\alpha\) on \((t_n-t_{cm})\) is approximated with sufficient accuracy by the theoretical equation (3). For comparison, on the same graph the dashed line shows the relation between \(\alpha\) and \((t_n-t_{cm})\), constructed from data of a series of experiments at pressures of \(0.9 \div 1.2\) atm [3].

Fig. 1. Dependence of the heat-transfer coefficient \(\alpha\) \([\mathrm{kcal}/\mathrm{m}^2\cdot\mathrm{h}\cdot\mathrm{deg}]\) on \(t_n-t_{cm}\);
\(\times\)—ortho-xylol; \(\square\)—aniline.

Aniline and orthoxylol, used as an ant wetting coating, caused unstable dropwise condensation. The drops were of increased size and, gradually spreading over the surface, gave a mixed type of condensation. In this case their shape resembled a truncated, compressed spheroid. The photographs (Fig. 3) show a pattern of dropwise condensation typical for aniline and fatty acids.

condensation. Figure 4 gives the results of experiments with aniline and o-xylene.

Conclusions

1. On the basis of the proposed hypothesis of heat transfer in dropwise condensation as a steady-state heat-conduction process, an equation has been obtained for the heat-transfer coefficient. In doing so, the parameters determining heat transfer have been identified. Investigation of droplet growth and comparison of the experimental data made it possible to obtain the design formula (3), which can be used in designing industrial apparatus.

2. An experimental study of dropwise condensation has been carried out, including measurement of the area occupied by droplets, the mean droplet size, and the wetting angle. It has been established that the type of hydrophobizing agent does not affect heat transfer. The exceptions are hydrophobizing agents that produce unstable dropwise condensation (aniline, o-xylene).

References

1. Ivanov V. V. Heat transfer in dropwise condensation of steam. Izv. TPI, vol. 110, 1962.

2. Kutateladze S. S. Heat Transfer in Condensation and Boiling. 1st and 2nd eds., Mashgiz, 1949 and 1952.

3. Kutateladze S. S., Borishanskii V. M., Novikov I. I., Fedyinskii O. S. Liquid-Metal Heat-Transfer Media. Atomizdat, 1958.

4. Sugawara S., Michiyoshi I., Mem. Fac. Eng., 18, No. 11, 1956.