Nusselts film condensation theory

When a vapour condenses on a vertical or inclined surface a liquid film develops, which flows downwards under the influence of gravity. When the vapour velocity is low and the liquid film is very thin, a laminar flow is created in the condensate. The heat will mainly be transferred by conduction from the surface of the condensate to the wall. The heat transferred by convection in the liquid film is negligibly small. 

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. 

As Fig. 4.6 shows, saturated steam at a temperature s is condensing on a vertical wall whose temperature 0 is constant and lower than the saturation temperature. A continuous condensate film develops which flows downwards under the influence of gravity, and has a thickness 5 x) that constantly increases. The velocity profile w(y), with w for wx, is obtained from a force balance. Under the assumption of steady flow, the force exerted by the shear stress are in equilibrium with the force of gravity, corresponding to the sketch on the right hand side of Fig. 4.6 

Considering only the vapour space, it is valid there that 

Under the assumption of temperature independent dynamic viscosity, (4.3) is transformed into 

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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... 

Nusselt s film condensation theory presumes a laminar film flow. As the amount of condensate increases downstream, the Reynolds number formed with the film thickness increases. The initially flat film becomes wavy and is eventually transformed from a laminar to a turbulent film the heat transfer is significantly better than in the laminar film. The heat transfer in turbulent film condensation was first calculated approximately by Grigull [4.14], who applied the Prandtl analogy for pipe flow to the turbulent condensate film. In addition to the quantities for laminar film condensation the Prandtl number appears as a new parameter. The results can not be represented explicitly. In order to obtain a clear representation, we will now define the Reynolds number of the condensate film... 

In the framework of Nusselt s film condensation theory, using the mass flow rate for the condensate, eq. (4.9),... 

By eliminating the film thickness 5 using a = AL/ , the heat transfer equation for Nusselt s film condensation theory can also be written as... 

The results of an analytical solution of the differential equations for a turbulent condensate film on a vertical tube are reproduced in Fig. 4.12, [4.15]. Line A represents Nusselt s film condensation theory according to (4.39). 

If a deviation of 1 % from the values for Nusselt s film condensation theory is permitted, this leads to a Reynolds number of... 

The factor / takes into account here the waviness of the laminar condensate film, / 1.15 QlalII is the heat transfer coefficient for laminar film condensation from Nusselt s film condensation theory and aturb is that for a turbulent condensate film, which is found, for example from (4.41) together with (4.42). 

Nusselt extended his film condensation theory to take into account the influence of vapour flowing along the condensate film on the velocity of the condensate. The boundary conditions for (4.6) are no longer dw/dy = 0 for y = 6, instead the velocity profile ends with a finite gradient at the free surface of the film corresponding to the shear stress exerted by the flowing vapour. In (4.6) for the velocity profile... 

Similarly to Nusselt s film condensation theory, in the condensation of vapour mixtures, the heat flux transferred increases with the driving temperature difference — 1 0. According to Nusselt s film condensation theory, the heat transfer coefficient decreases with the driving temperature difference according to a ( oo- o) 1/4 (4-12). The heat flux increases in accordance with q co — q)3/4. Fig. 4.21a shows clearly that a minimum for the transferred heat flux exists at a certain temperature oa. This is because the temperature difference dj — r)(j between the condensate surface and the wall, which is decisive for heat transfer, also assumes a minimum this can be explained by the boiling diagram, Fig. 4.21. 

We will presume a sufficiently large temperature difference dd — dd0, such that the vapour in its initial state A condenses, and a condensate accumulates. This is indicated by point B in Fig. 4.21b. The temperature at the phase interface is then equal to the boiling point of the liquid mixture and the composition of the accumulated condensate is identical to that of the vapour. This is known as local total condensation. The wall temperature, which is assumed to be constant, is characterised by point C. The line BC corresponds to the temperature difference —dd0 that is decisive for the heat flux q. If Nusselt s film condensation theory was also valid for vapour mixtures, then q — dd0)3/4. If dd — dd0 is kept constant... 

With these estimated values the vapour side heat transfer coefficient aG and the mass transfer coefficient [3G can be calculated. Just as the heat transfer coefficient aL of the condensate film is also known, which in laminar film condensation is yielded from Nusselt s film condensation theory (4.39), and for turbulent film condensation from (4.41). From (4.67) the temperature... 

According to Nusselt s film condensation theory, from (4.13a), under the assumption QG Qi, the heat transfer coefficient follows as... 

If the wall is inclined at an angle 7 to the vertical then the acceleration due to gravity g has to be replaced by its component parallel to the wall g cos 7 with 0 < 7 < 7r/2. The equation also holds for condensation of quiescent vapours on the inside or outside of a vertical tube, if the diameter of the tube is large in comparison to the film thickness. The width b has to be replaced by b = ir d. If a deviation of 1 % from the values from Nusselt s film condensation theory is permitted, then the equation is valid up to a Reynolds number... 

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). 

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. 

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 ... 

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. 

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).]... 

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... 

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