Nusselt theory

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 ... 

Kirkbride (K17), 1934 Flow of water and 4 oils outside tubes, JVro = 0.04-2000. Film thicknesses (maximum wave heights) measured by micrometer. Wavy flow is described, and corrections to Nusselt theory derived for heat transfer in laminar wavy film flow. 

Labuntsov (LI), 1956 Heat transfer to falling films (laminar flow) effects of convective heat transfer and inertia forces (neglected in Nusselt theory) considered experimentally and theoretically. 

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 ... 

With the above corrections, the presence of waves can easily explain the noted 15-20 percent discrepancy between the Nusselt theory and experimental data. 

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)... 

In terms of the local heat transfer coefficient in the Mh row of the bundle, the Nusselt theory predicts ... 

Beatty and Katz [92] completely neglected surface tension effects and arrived at a very simple expression for a gravity-drainage heat transfer coefficient using 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. 

It must be noted that predictions of heat transfer coefficients in all mentioned situations may be treated, as a rule, as conservative as long as the correlation is based on the Nusselt theory. Two important additional phenomena, though, are not included vapor superheat and vapor shear effects. The influence of superheating can be included (although the effect is usually small) by the sixth correlation from the top in Table 17.23. 

Horizontal Surfaces. For annular film flow, the Nusselt-theory-based correlations usually fail to provide acceptable predictions. This type of flow is shear-dominated flow. This problem has been a subject of extensive research, and numerous correlations can be found in literature [78]. The correlation given by Shah [88] in Table 17.25 is the best, as it is valid for a wide range of fluids and flow conditions. The mean deviation for 474 data points analyzed was found to be 15 percent. 

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 ... 

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