On Heat Transfer During Dropwise Condensation of Vapor
Abstract Generated abstract
This paper investigates heat transfer during dropwise condensation of slightly superheated steam on a brass tube at atmospheric pressure, addressing conditions closer to industrial heat exchangers than earlier studies on copper or chromium-plated surfaces. Experiments varied cooling-water velocity, heat load, temperature head, and hydrophobizing agents, with heat-transfer coefficients calculated independently from condensate production and cooling-water heating. The results indicate that, when stable dropwise condensation is maintained, the type of promoter has little practical effect on heat transfer, while the overall heat-transfer coefficient increases with cooling-water velocity and with the temperature head between steam and the cooled wall. The paper relates these trends to a heat-transfer model and presents the data as a basis for estimating coefficients when velocity, heat load, or temperature head is known.
Full Text
On Heat Transfer During Dropwise Condensation of Steam
V. V. Ivanov
(Presented by Prof. G. I. Fuks, Doctor of Technical Sciences)
Heat transfer in condensers, evaporators, boilers, and in a number of other heat-exchange apparatuses is substantially increased if dropwise condensation occurs on the cooling surfaces. In this case, a large part of the wall surface is accessible for direct contact with the steam. The increased heat-transfer coefficients from the steam to the “bare” portions of the wall between droplets contribute to an increase in the overall heat-transfer coefficient from the steam to the cooling water.
A number of works [1, 2, 3] have been devoted to the study of heat transfer during dropwise condensation. These works are experimental in character and contain a qualitative analysis of the experimental data. The experiments that were carried out made it possible to establish the order of magnitude of the heat-transfer coefficient and certain regularities in its variation as a function of the velocity of the cooling water, the heat load, etc. These experiments were conducted over a small range of variation of the temperature head between the steam and the cooled wall, and with a small number of hydrophobizing agents. Dropwise condensation occurred on copper or chromium-plated tubes, which are little used in industrial heat-exchange apparatuses.
Below are presented the results of an investigation of the heat-transfer coefficient during dropwise condensation, obtained on an experimental installation, a detailed description of which is given in [4]. Dropwise condensation occurred on a brass tube \((\lambda = 105\ \mathrm{W}/\mathrm{m}\cdot{}^\circ\mathrm{K})\) with a wall thickness of \(0.001\ \mathrm{m}\). The experiments were carried out at atmospheric pressure; the steam had slight superheat \((5\text{–}20^\circ\mathrm{K})\). The flow rate of cooling water in the experiments varied from \(70\) to \(850\ \mathrm{kg}/\mathrm{h}\), which corresponded to a change in the velocity of the cooling water from \(0.01\) to \(1.3\ \mathrm{m}/\mathrm{s}\).
Kerosene, machine oil, oleic and stearic acids dissolved in ether were used as hydrophobizing agents. The temperature head between the steam and the cooling surface in the experiments varied over wide limits: from \(1\) to \(20^\circ\mathrm{K}\).
In carrying out the experiments, the value of the heat-transfer coefficient was found from the relation
\[ K=\frac{G_k(i_{\mathrm{п}}-i_k)}{3600F(t_{\mathrm{н}}-t_{\mathrm{в}})} \left[\frac{\mathrm{W}}{\mathrm{m}^2\cdot{}^\circ\mathrm{K}}\right], \]
where \(G_k\) is the hourly condensate flow rate; \(F\) is the condensation surface, equal to \(0.0226\ \text{m}^2\); \((i_p-i_k)\) is the mean difference in heat contents of the steam and condensate at the inlet to and outlet from the condenser; \((t_n-t_v)\) is the temperature difference between the condensing steam and the cooling water.
At the same time, for checking purposes, the heat-transfer coefficient \(K\) was calculated from the relation
\[ K=\frac{G_v\left(t_{\mathrm{out}}-t_{\mathrm{in}}\right)} {3600\,F\,(t_n-t_v)} \left[ \frac{\text{W}}{\text{m}^2\cdot{}^\circ\!K} \right], \]
where \(G_v\) is the hourly flow rate of cooling water, and \((t_{\mathrm{out}}-t_{\mathrm{in}})\) is the mean temperature difference of the cooling water at the outlet and inlet of the condenser.
As the experiments showed, the difference in the values of \(K\) determined by these methods did not exceed 8%.
Figure 1 presents, in logarithmic coordinates, the experimental dependence of the heat-transfer coefficient \(K\) on the velocity of the cooling water \(W_0\) \([\text{m/sec}]\), showing the nature of the increase of \(K\) with increasing \(W_0\). A comparison of the data presented shows that the type of promoter producing stable dropwise condensation has practically no effect on heat transfer. Therefore, in the subsequent processing of the experimental results no distinction is made among the data obtained with kerosene, machine oil, oleic acid, and stearic acid.
Fig. 1. Dependence of the heat-transfer coefficient \(K\) \([\text{W}/\text{m}^2K]\) on the velocity of the cooling water \(W_0\) \([\text{m/sec}]\).
○ — kerosene, ● — oleic acid, —▼ machine oil SU, ▽ — stearic acid dissolved in ether.
Figure 2 presents experimental data showing the dependence of the heat-transfer coefficient \(K\) on the heat load of the cooling surface.
An attempt was made to establish the dependence of the heat-transfer coefficient on the temperature head between the steam and the cooled wall, a quantity usually specified in calculating heat transfer (Fig. 3).
Fig. 2. Dependence of the heat-transfer coefficient \(K\) \([\mathrm{W}/\mathrm{m}^2\,^\circ\mathrm{K}]\) on the heat load of the cooling surface \(q_k\) \([\mathrm{W}/\mathrm{m}^2]\).
Despite the considerable scatter of the experimental points, it may be noted that, under conditions of stable dropwise condensation, the value of \(K\) (in contrast to the heat-transfer coefficient) increases as \((t_n - t_{\mathrm{st}})\) increases. This effect is consistent with the equations obtained in [4]. Indeed, the heat-transfer coefficient \(K\) can be determined as
\[ K = \frac{\bar{\alpha}\cdot K_0}{\bar{\alpha}+K_0}, \]
where \(\bar{\alpha}\) is the heat-transfer coefficient on the steam side; \(K_0\) is the heat-transfer coefficient through the wall to the cooling water. The value of the heat-transfer coefficient \(\bar{\alpha}\) can be calculated from the equation [4]
\[ \bar{\alpha}=\frac{C(p)\gamma r}{2(t_{\mathrm{n}}-t_{\mathrm{st}})}+1.22K_0, \]
in which \(C(p)\) is a constant \((\mathrm{m}/\mathrm{s})\) depending on the pressure; \(\gamma\) is the specific weight of the condensate, and \(r\) is the latent heat of vaporization.
Combining the last two formulas, we obtain an equation for calculating \(K\)
\[ K=K_0\left[ \frac{0.5C(p)\gamma r+1.22K_0(t_{\mathrm{n}}-t_{\mathrm{st}})} {0.5C(p)\gamma r+2.22K_0(t_{\mathrm{n}}-t_{\mathrm{st}})} \right]. \]
At constant steam pressure and unchanged temperature \(t_{\mathrm{in}}\), an increase in the temperature head \((t_{\mathrm{n}}-t_{\mathrm{st}})\) is determined mainly by an increase in the cooling-water velocity \(W_0\). Together with the velocity \(W_0\), the heat-transfer coefficient through the wall to the cooling water, \(K_0\), also increases.
Fig. 3. Relationship of the heat-transfer coefficient \(K\) \([\mathrm{W}/\mathrm{m}^2\,^\circ K]\) with the temperature head between the steam and the cooled wall \((t_{\mathrm{n}}-t_{\mathrm{st}})^\circ K\).
○ — kerosene; ● — oleic acid; ▼ — machine oil SU; ▽ — stearic acid dissolved in ether.
As the processing of the experimental data shows, with an increase in \(W_0\) and, consequently, in \((t_{\mathrm{n}}-t_{\mathrm{st}})\), the value \(K_0\) increases more strongly than the expression in square brackets decreases. This ultimately leads to an increase in \(K\).
The experimental data obtained can be used for a direct estimate of the heat-transfer coefficient \(K\), if \(W_0\), \(q_k\), or \((t_{\mathrm{n}}-t_{\mathrm{st}})\) are known.
REFERENCES
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- Fitzpatrick J., Baum S. W. Mc Adams. Trans. Am. Inst. Chem. Engrs., 35, 1939.
- Ivanov V. V. Study of heat exchange during dropwise condensation of steam. Proceedings of the TPI, vol. 110, 1962.