Nusselt theory, condensation

Dukler Theory The preceding expressions for condensation are based on the classical Nusselt theoiy. It is generally known and conceded that the film coefficients for steam and organic vapors calculated by the Nusselt theory are conservatively low. Dukler [Chem. Eng. Prog., 55, 62 (1959)] developed equations for velocity and temperature distribution in thin films on vertical walls based on expressions of Deissler (NACA Tech. Notes 2129, 1950 2138, 1952 3145, 1959) for the eddy viscosity and thermal conductivity near the solid boundaiy. According to the Dukler theoiy, three fixed factors must be known to estabhsh the value of the average film coefficient the terminal Reynolds number, the Prandtl number of the condensed phase, and a dimensionless group defined as follows ... 

FIG. 5-8 Dukler plot showing average condensing-film coefficient as a function of physical properties of the condensate film and the terminal Reynolds number. (Dotted line indicates Nusselt theory for Reynolds number < 2100.) [Reproduced hy permission from Chem. Eng. Prog., 55, 64 (1959).]... 

Figure 11.4 Film wise condensation departure from the Nusselt theory for liquid metals.

Vertical tubes. In film-type condensation, the Nusselt theory shows that the condensate film starts to form at the top of the tube and that the thickness of the film increases rapidly near the top of the tube and then more and more slowly in the remaining length. The heat is assumed to flow through the condensate film solely by conduction, and the local coefficient is therefore given by... 

Shang and Adamek [15] recently studied laminar film condensation of saturated steam on a vertical flat plate using variable thermophysical properties and found that the Nusselt theory with the Drew [14] reference temperature cited above produces a heat transfer coefficient that is as much as 5.1 percent lower than their more correct model predicts (i.e., the Nusselt theory is conservative). 

Condensate Waves and Turbulence. As the local condensate film thickness (i.e., the film Reynolds number Rez) increases, the film will become unstable, and waves will begin to grow rapidly. This occurs for Re, > 30. Kapitza [16] has shown that, in this situation, the average film thickness is less than predicted by the Nusselt theory and the heat transfer coefficient increases accordingly. Kutateladze [17] therefore recommends that the following correction be applied to Eq. 14.12 ... 

Equation 14.56 is compared to data from 12 investigations, using four different fluids, in Fig. 14.12. Equation 14 54 from Nusselt theory is also shown for comparison. It is clear that Eq. 14.56 approaches the Nusselt result at low vapor velocities (Fd —> °°) and gives a reasonable value of the average condensation heat transfer coefficient (greater than the Nusselt theory)... 

Heat Transfer Correlations for External Condensation. Although the complexity of condensation heat transfer phenomena prevents a rigorous theoretical analysis, an external condensation for some simple situations and geometric configurations has been the subject of a mathematical modeling. The famous pioneering Nusselt theory of film condensation had led to a simple correlation for the determination of a heat transfer coefficient under conditions of gravity-controlled, laminar, wave-free condensation of a pure vapor on a vertical surface (either flat or tube). Modified versions of Nusselt s theory and further empirical studies have produced a list of many correlations, some of which are compiled in Table 17.23. 

A presence of interfacial waves increases the heat transfer coefficient predicted by Nusselt theory by a factor up to 1.1. An underprediction of a heat transfer coefficient by the Nusselt theory is more pronounced for larger condensate flow rates. For laminar condensation having both a wave-free and wavy portion of the condensate film, the correlation based on the work of Kutateladze as reported in [81] (the fourth correlation from the top of Table 17.23) can be used as long as the flow is laminar. 

In stratified flow, the stratified layer at the lower part of the tube free-flow area is influenced primarily by shear effects, while a thin film covers the upper portions of the inner tube wall and stratifies under the influence of gravity. The heat transfer conditions in two regions are quite different, but it is a standard practice to correlate heat transfer based on the entire perimeter. In Table 17.25, a correlation based on the modified Nusselt theory is given for stratified flow, developed by Chato [87] and modified by Jaster and Kosky, as reported by Carey [76]. Consult Carey [76] and Butterworth [81] for a detailed analysis of related phenomena. The most recent condensation correlations are given by Dobson and Chato [89]. 

Because this type of operation essentially involves the Nusselt theory with enhanced liquid flow, a heat transfer coefficient approach is used for calculative purposes, rather than transfer units. Temperature differences are calculated as if a surface condenser were being used. If a single-component vapor is condensing at a constant pressure, the temperature difference at the liquid coolant inlet is T — q. Similarly, the temperature difference at the liquid coolant outlet is Ti — to- This is true because the vapor temperature remains constant from the first drop of condensate to the last drop. Thus, the logarithmic mean temperature driving force is ... 

When considering commercial equipment, there are several factors which prevent the true conditions of Nusselt s theory being met. The temperature of the tube wall will not be constant, and for a vertical condenser with a ratio of AT at the bottom to AT at the top of five, the film coefficient should be increased by about 15 per cent. 

Using an analysis similar to Nusselt s theory on filmwisc condensation presented in the next section, Bromley developed a theory for the prediction of heat flux for stable film boiling on the outside of a horizontal cylinder. The heat Ilux for film boiling on a horizontal cylinder or sphere of diameter D is given by... 

In 1916, Nusselt [4.2] had already put forward a simple theory for the calculation of heat transfer in laminar film condensation in tubes and on vertical or inclined walls. This theory is known in technical literature as Nusselt s film condensation theory. It shall be explained in the following, using the example of condensation on a vertical wall. 

Deviations from Nusselt s film condensation theory... 

Experiments on film condensation of saturated vapour on vertical walls yield deviations by as much as +25% from the heat transfer coefficients according to Nusselt s film condensation theory. There are different reasons that are decisive for this. 

Nusselt s film condensation theory presumes an even increase in the thickness of the film due to further condensation. However experiments, among others [4.4] to [4.6], have shown that even in a flow that is clearly laminar, waves can develop at the film surface. These types of waves were not only observed on rough but also on polished surfaces. Obviously this means that the disturbances in the velocity that are always present in a stream are not damped under certain conditions, and so waves form. They lead to an improvement in the heat transfer of 10 to 25 % compared to the predictions from Nusselt s theory. According to Grimley [4.7], waves and ripples appear above a critical Reynolds number... 

In (4.12) and (4.13) of Nusselt s film condensation theory the material properties of the condensate are presumed to be independent of temperature. This assumption is well met if the temperature drop ds — -do in the condensate film is sufficiently small. In any other case the change in the dynamic viscosity, the thermal conductivity, and on a lower scale the density of the condensate film with the temperature has to be considered. In place of (4.5) the momentum balance appears... 

With the additional assumption that the density of the liquid film only slightly depends on the temperature, and that it is much larger than that of the vapour, gi, qg, we obtain the result that the relationship between the heat transfer coefficient a, and the heat transfer coefficient ocnu, according to Nusselt s film condensation theory, can be represented by... 

The index s signifies the material properties of the condensate film at the saturation temperature, whilst index 0, indicates those properties that are formed at the wall temperature. The heat transfer coefficient QtNu according to Nusselt s film condensation theory, (4.12), is calculated with the mean material properties... 

Fig. 4.7 illustrates (4.24). As we can see from the graph, the temperature dependence of the dynamic viscosity and the thermal conductivity can have a marked influence on the heat transfer, as far as they change starkly with the temperature. In condensing steam, for a temperature difference between the saturation and wall temperatures of s — i o < 50 K, the material properties vary for As/Ao between 0.6 and 1.2 and for Vs/vo between 1 and 1.3. This region is hatched in Fig. 4.7. It is clear that within this region the deviations from Nusselt s film condensation theory are less than 3%.

It follows from this that, under the assumptions made, the specific enthalpy hi, of the flowing condensate is independent of the film thickness. The equation further shows that the enthalpy of vaporization Ahv in the equations for Nusselt s film condensation theory has to be replaced by the enthalpy difference Ah. If we additionally consider that the temperature profile in the condensate film is slightly curved, then according to Rohsenow [4.10] in place of (4.27), we obtain for Ah the more exact value... 

A further, and likewise small, deviation from Nusselt s film condensation theory is found for superheated vapour. In addition to the enthalpy of vaporization, the superheat enthalpy CpQ ( g — < s) has to be removed in order to cool the superheated vapour from a temperature da to the saturation temperature t9s at the phase interface. Instead of the enthalpy difference Ahv according to (4.28), the enthalpy difference... 

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