Heat Transfer During Dropwise Condensation of Vapor
Abstract Generated abstract
This paper develops a theoretical and experimental treatment of heat transfer during dropwise condensation on nonwettable cooled surfaces. Modeling the process as steady heat conduction through condensate drops and the wall, it derives expressions for the heat-transfer coefficient in terms of drop-covered area, average drop radius, contact angle, condensate thermal conductivity, and wall-side heat transfer. The analysis includes a solution for the temperature field and heat flow in a spherical-segment drop and uses photographic measurements of condensing surfaces to estimate the required geometric parameters. The resulting approximate formula predicts heat-transfer coefficients of the same order as observed experimentally and helps explain their relative independence from heat load over the studied range.
Full Text
Heat Transfer in Dropwise Condensation of Vapor
V. V. Ivanov
(Presented by Professor Doctor G. I. Fuks)
One of the ways of intensifying heat transfer in heat-exchange apparatuses may be the artificial production of dropwise condensation. This form of condensation is accompanied by very high heat-transfer coefficients (up to 200000 kcal/m²·h·°C) and, consequently, makes it possible, without changing the thermal load, to reduce the cooling surfaces. The use of heat-resistant polymeric hydrophobizing agents insoluble in the condensate opens broad possibilities for the practical use of dropwise condensation. Thus, Kalberg and Kendall [1], using a silicone varnish as the hydrophobizing agent, obtained a coating that retained its original properties after 1500 hours of operation.
An important problem is the creation of a theory of heat transfer in dropwise condensation that would make it possible to analyze the processes occurring in heat exchangers with a nonwettable surface, to design such apparatuses, and to accelerate their introduction into industry. In addition, the study of heat transfer in dropwise condensation is necessary for the design of heat exchangers of mercury-power installations, where this process takes place [8]. Although the mechanism of dropwise condensation has at present been studied rather well [2, 3, 4], attempts to calculate heat transfer theoretically [5, 6] have not led to satisfactory results. The equations obtained do not make it possible to explain the entire body of reliable experimental data and cannot serve as a dependable basis for practical calculations.
In the present work the results of theoretical and experimental investigations of heat exchange in dropwise condensation are presented.
Derivation of the equation for the heat-transfer coefficient in dropwise condensation
We shall proceed from the following assumptions:
- Heat transfer in dropwise condensation is regarded as a process of heat conduction through the drops and the cooled wall. In this case there is no thermal resistance at the vapor—drop surface boundary, and the surface of the drops itself has the saturation temperature.
II. The process is stationary, more precisely, averaged over the time under consideration: (a) the temperature field is assumed to be established and unchanging in time; (b) the effect of pulsations associated with the periodicity of the process of drop formation and detachment is not taken into account, since the pulsating regime is replaced by a stationary model.
Then the time-averaged over time \(\tau\) heat-transfer coefficient in dropwise condensation is determined by the equation
\[ \alpha=\frac{Q}{(t_n-t_{st})F\tau}, \tag{1} \]
where \(Q/\tau\) is the heat flux from the vapor to the cooling water through the condensation surface \(F\); \((t_n-t_{st})\) is the average, over time \(\tau\), temperature difference from the saturated vapor to the wall.
Fig. 1. Dropwise condensation on a cooled wall.
If the area of the surface covered by drops is \(F_k\), then, in accordance with Fig. 1, the quantity \(Q/\tau\) can be expressed as
\[ \frac{Q}{\tau} = K_{pk}(t_n-t_v)F_k + K_0(t_n-t_v)(F-F_k). \tag{2} \]
Here \(K_{pk}\) is the heat-transfer coefficient through the drops and the wall beneath them to the cooling water; \(K_0\) is the heat-transfer coefficient from the vapor to the “bare” wall and through the wall to the cooling water; \((t_n-t_v)\) is the average temperature difference between the vapor and the cooling water.
Starting from the relation of thermal resistances, we have
\[ \frac{1}{K_{pk}} = \frac{1}{K_k} + \frac{1}{K}; \tag{3} \]
\[ \frac{1}{K_0} = \frac{1}{\alpha_p} + \frac{1}{K}. \tag{4} \]
In these formulas \(K_k\) is the heat-transfer coefficient through the drops; \(\alpha_p\) is the heat-transfer coefficient from the vapor to the “bare” surface; \(K\) is the heat-transfer coefficient through the wall to the cooling water, with
\[ \frac{1}{K}=\frac{\delta_{st}}{\lambda_{st}}+\frac{1}{\alpha_v}, \]
where \(\alpha_v\) is the heat-transfer coefficient from the wall to the cooling water.
Combining (1), (2), (3), and (4), we obtain
\[ \alpha(t_n-t_{st})F = \left[ K(t_n-t_{st})-K(t_{st}-t_v) \right] \left[ \frac{K_k F_k}{K+K_k} + \frac{\alpha_p(F-F_k)}{K+\alpha_p} \right]. \]
Replacing \(-K(t_{cm}-t_v)\) by \(-\bar{\alpha}(t_n-t_{cm})\), we solve the last equation for \(\bar{z}\):
\[ \bar{z}= \frac{ \dfrac{K_k F_k}{K+K_k} - \dfrac{F-F_k}{1+\dfrac{K}{\alpha_p}} }{ \dfrac{F_k}{K+K_k} + \dfrac{F-F_k}{K+\alpha_p} } \tag{5} \]
Let us now consider two special cases of formula (5).
I. \(F_k=0\), and \(\bar{\alpha}=\alpha_p\). Physically this means that condensation does not occur, but only cooling of superheated vapor, and the heat-transfer coefficient \(\alpha_p\) can be calculated from the equations of convective heat exchange.
II. \(F_k=F\), and \(\bar{\alpha}=K_k\). In this case the individual drops, merging together, form a “single drop”—a film that completely covers the cooling surface, and dropwise condensation changes into film condensation.
From a practical point of view, the study of heat transfer in dropwise condensation of pure vapor is of great interest, since this may occur in heat exchangers operating not under vacuum but at excess pressure. Under these conditions, air leakage is eliminated, and the thermal resistance of the vapor-gas layer at the cooling surface ceases to be a significant factor: \(\dfrac{1}{\alpha_p}\approx 0\) and \(\alpha_p\to\infty\). The heat-transfer coefficient for dropwise condensation of pure vapor is found from (5):
\[ \bar{\alpha}_{\alpha_p\to\infty} = \frac{F}{F_k}\left(K_k+K\right)-K . \tag{6} \]
The quantity \(K_k\) can be determined from the relation
\[ K_k=\frac{Q_k}{F_k(t_n-t_k)}, \tag{7} \]
where \(Q_k\) is the heat flux passing through the drops, and \((t_n-t_k)\) is the mean temperature difference between the surface of the drops and their base.
It can be shown that the temperature field inside an individual drop having the shape of a spherical segment of radius \(R\) (Fig. 2) is given by the following expression:
\[ \frac{t(r,\Theta)-t_k}{t_n-t_k} \]
\[ = \frac{ \displaystyle \sum_{n=1}^{\infty} \left[ P_{n-1}(\cos\beta)-P_{n+1}(\cos\beta) \right] \left[ \left(\frac{r}{R}\right)^n P_n(\cos\Theta)-(\cos\beta)^n(-1)^n \right] }{ \displaystyle 1+\cos\beta - \sum_{n=1}^{\infty} \left[ P_{n-1}(\cos\beta)-P_{n+1}(\cos\beta) \right] (\cos\beta)^n(-1)^n } \tag{8} \]
Here \(t_n\) is the temperature of the spherical surface, \(t_k\) is the temperature of the base of the drop, \(\beta\) is the wetting angle, \(P_n(x)\) are Legendre polynomials of order \(n\), and \(r\) and \(\Theta\) are the current coordinates. For the particular case of a hemisphere \(\left(\beta=\frac{\pi}{2}\right)\), equation (8) takes the form
\[ \frac{t(r,\Theta)-t_k}{t_n-t_k} = \sum_{n=0}^{\infty}(-1)^n \frac{1\cdot 3\cdot 5\cdots(2n-1)(4n+3)} {2\cdot 4\cdot 6\cdots(2n+2)} \left(\frac{r}{R}\right)^{2n+1} P_{2n+1}(\cos\Theta), \]
which corresponds to the solution given in [9].
Fig. 2. Isotherms inside a spherical drop.
The heat through the spherical segment under consideration can be calculated with the aid of the temperature gradient at the surface
\[ Q=-\lambda \int_{0}^{\beta} \left(\frac{\partial t}{\partial r}\right)_{r=R} 2\pi R^2\sin\Theta\,d\Theta, \]
where \(\lambda\) is the thermal conductivity of the condensate.
Finding the temperature gradient from equation (8), we shall have
\[ Q = 2\pi\lambda(t_n-t_k)R\, \frac{ \displaystyle \sum_{n=1}^{\infty} \left[ P_{n-1}(\cos\beta)-P_{n+1}(\cos\beta) \right]\,n }{ \displaystyle 1-\cos\beta - \sum_{n=1}^{\infty} \left[ P_{n-1}(\cos\beta)-P_{n+1}(\cos\beta) \right] (\cos\beta)^n(-1)^n } \cdot \int_{\cos\beta}^{1} P_n(\cos\Theta)\,d(\cos\Theta) \]
The integral in the last equality is equal to [9]
\[ \int_{\cos\beta}^{1} P_n(\cos\Theta)\,d(\cos\Theta) = \frac{1}{2n+1} \left[ P_{n-1}(\cos\beta)-P_{n+1}(\cos\beta) \right]. \]
Therefore
\[ \frac{Q}{2\pi\lambda (t_h-t_c)R} = \frac{ \displaystyle \sum_{n=1}^{\infty} \frac{n}{2n+1} \left[ P_{n-1}(\cos \xi)-P_{n+1}(\cos \xi) \right]^2 }{ \displaystyle 1+\cos\beta - \sum_{n=1}^{\infty} \left[ P_{n-1}(\cos\beta)-P_{n+1}(\cos\beta) \right] (\cos\beta)^n(-1)^n }. \tag{9} \]
Substituting various values of the angle \(\beta\) into (9), we obtain
\[ Q=\pi\lambda(t_h-t_c)R\psi(\beta), \tag{10} \]
where \(\psi(\beta)\) is a dimensionless function of the contact angle, which is found from the curve (Fig. 3).
Fig. 3. Graphs of the functions \(\psi(\beta)\) and \(f(\beta)\).
The heat-transfer coefficient through a single drop is determined as
\[ K_{\mathrm{single}}=\frac{Q}{F_0(t_h-t_c)} . \tag{11} \]
Here \(F_0=\pi R_0^2\) is the area of the base of the drop, with \(R_0=R\sin\beta\) (Fig. 2). Combining (10) and (11), we obtain
\[ K_{\mathrm{single}} = \frac{\lambda\psi(\beta)}{R_0\sin\beta} = \frac{\lambda f(\beta)}{R_0}. \tag{12} \]
The function
\[ f(\beta)=\frac{\psi(\beta)}{\sin\beta} \]
is presented in Fig. 3.
If \(N\) drops of different sizes have condensed on a nonwettable cooled wall, then the heat flux through them is equal to
\[ Q_K=Q_1+Q_2+\cdots+Q_N, \]
where \(Q_1\) is the heat flux through a drop of radius \(R_1\), \(Q_2\) is the heat flux through a drop of radius \(R_2\), and so on. As direct processing of photographs of the process of dropwise condensation shows, the surface-
…let the surface of the wall be covered with drops of various sizes, but approximately identical in shape ($\beta = idem$).
Therefore
\[ Q_k = \pi\lambda (t_n-t_k)\psi(\beta) \left(\frac{R_1+R_2+\cdots+R_N}{N}\right)N = \pi\lambda (t_n-t_k)\psi(\beta)R_{\mathrm{av}}N \tag{13} \]
Comparing now (6), (7), and (13), we obtain
\[ \alpha = \frac{F}{F_k} \left[ \frac{\pi\lambda \psi(\beta)R_{\mathrm{av}}N}{F_k} + K \right] - K . \]
The latter relation can be considerably simplified if one sets $F_k=N\pi R_{\mathrm{av}}^2$. The physical meaning of such an assumption consists in replacing the various drops located on the surface by identical drops of average size. In this case the calculated and actual values of $F_k$ will be the closer, the less the sizes of the individual drops differ from one another and the greater their number. Taking this into account, we obtain the following expression for $\alpha$:
\[ \alpha = \frac{F}{F_k} \left[ \frac{\lambda\psi(\beta)}{R_{\mathrm{av}}} + K \right] - K . \tag{14} \]
Equation (14) makes it possible to reveal certain essential features of the mechanism of heat transfer in dropwise condensation. It also makes it possible to estimate the values of the heat-transfer coefficient $\alpha$ through quantities that permit direct measurement: $F_k$, $\beta$, and $R_{\mathrm{av}}$. For this purpose, a large number of photographs of the process of dropwise condensation were processed. The calculation of $F_k$ and $R_{\mathrm{av}}$ was carried out on a flat cooled wall coated with a certain hydrophobizing agent. A linear scale was applied to the wall (1 division visible in the photographs corresponds to 5 mm).
As it turned out, the area occupied by drops and the average drop size, over wide limits, do not depend on the heat load of the cooling surface—$q_k \left[\dfrac{\mathrm{kcal}}{\mathrm{m}^2\mathrm{h}}\right]$.
The experiments performed showed the approximate constancy of $\alpha$ over a wide interval of variation of $q_k$, which confirms the results previously obtained by some investigators [7]. Qualitatively, this effect agrees well with equation (14). Indeed, if $F_k$, $R_{\mathrm{av}}$, and $\beta$ do not depend on the heat load, then the influence of $q_k$ on $\alpha$ is manifested only through the quantity $K$. Comparison of experimental data shows that, with increasing $q_k$, the coefficient $K$ increases approximately from 2800 to 5500 $\mathrm{kcal}/\mathrm{m}^2\mathrm{h}\cdot{}^\circ\mathrm{C}$, whereas $\alpha$ in the region studied varies within the limits from 20,000 to 200,000 $\mathrm{kcal}/\mathrm{m}^2\mathrm{h}\cdot{}^\circ\mathrm{C}$. As a result, the quantity $K$ gives an insignificant correction to equation (14), and the heat-transfer coefficient in dropwise condensation, according to (14), is practically independent of $q_k$.
To calculate heat transfer it is necessary to know the relation of the area occupied by drops and the average drop size to the temperature difference between the vapor and the cooled wall. The solution of this question is complicated by the fact that, for each value of $F_k$ and $R_{\mathrm{av}}$, there is its own temperature dependence, not amenable to theoretical calculation. Under these conditions, the only reliable method for determining $F_k$, $R_{\mathrm{av}}$, and $\beta$ is experiment. Analysis of motion-picture frames of the process of dropwise…
а
б
Fig. 4. Photographs of the process of dropwise condensation on a vertical cooled wall coated with kerosene.
...of dropwise condensation made it possible to establish a certain influence of the temperature head on \(F_k\) and \(R_{ocr}\). The experiments were carried out over a wide range of variation of \((t_n - t_{st})\): from 0.8 to 20°C.
Quantitative processing of greatly enlarged photographs of the dropwise-condensation process showed that \(F_k\) in the region studied constitutes \(\simeq (35 \div 55)\%\) of the entire cooling surface, \(R_{ocr}\) is \((0.1 \div 0.5)\) mm, and the angle \(\beta\) for oleic and stearic acids, kerosene, and SU machine oil varies approximately from 90 to 100°. As an approximation for calculations one may take \(\beta \simeq 95^\circ\), especially since the function \(f(\beta)\) entering into (14) changes very little near the point \(\beta = 95^\circ\) (Fig. 3).
Substitution of these values of \(F_k\), \(R_{ocr}\), and \(\beta\) into equation (14) gives the limits of variation of \(\bar{\alpha}\) with increasing \((t_n - t_{st})\), from 135000 to 20000 kcal/m²·h·°C.
The values obtained for \(\bar{\alpha}\) are, in general, in satisfactory agreement with the experimental data. Therefore relation (14), as a first approximation, may be used to estimate the heat-transfer coefficient in dropwise condensation, if \(F_k\), \(R_{ocr}\), and \(\beta\) are known from experiment.
REFERENCES
- Kullberg G., Kendall H., Chem. Eng. Prog., 56, No. 1, 1960.
- Emmons H., Trans. Am. Inst. Chem. Eng., 35, 1939.
- Hampson H., Engineering, 172, No. 4461, 1951.
- Hampson H., Engineering, 179, No. 4655, 1955.
- Fatica N., Katz D., Chem. Eng. Prog., 45, No. 11, 1949.
- Sugawara S., Michiyoshi J., Mem. Fac. Eng., 18, No. 11, 1956.
- Gröber H., Erk S., and Grigull U. Fundamentals of the Theory of Heat Transfer. IIL, 1958.
- Lozhkin A. N., Kanaev A. A. Binary Installations. Mashgiz, 1946.
- Koshlyakov N. S. The Basic Differential Equations of Mathematical Physics. ONTI, 1936.