Nusselt equation for condensation

The equation given by Bromley (1950) can be used to estimate the heat-transfer coefficient for film boiling on tubes. Heat transfer in the film-boiling region will be controlled by conduction through the film of vapour, and Bromley s equation is similar to the Nusselt equation for condensation, where conduction is occurring through the film of condensate. 

Combining Equations 15.86 and 15.88 gives the Nusselt Equation for condensation inside vertical tubes ... 

At low flow rates, a vapor may be condensed inside a horizontal or nearly horizontal tube under the condition that the condensate flhn drains down the inside surface of the tube by gravity and in laminar flow into the liquid pool at the bottom of the tube and then out the tube outlet. Kern [38] proposed a simple modification of the Nusselt equation for condensation outside horizontal tubes (see next section) that can be used for this case ... 

Vertical Tubes For the following cases Reynolds number < 2100 and is calculated by using F = Wp/ KD. The Nusselt equation for the heat-transfer coefficient for condensate films may be written in the following ways (using liquid physical properties and where L is the cooled lengm and At is — t,) ... 

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 ... 

For liorizontal in-tube condensation at low flow rates Kern s modification (Process Heat Transfer, McGraw-Hill, New York, 1950) of the Nusselt equation is vahd ... 

The basic equations for filmwise condensation were derived by Nusselt (1916), and his equations form the basis for practical condenser design. The basic Nusselt equations are derived in Volume 1, Chapter 9. In the Nusselt model of condensation laminar flow is assumed in the film, and heat transfer is assumed to take place entirely by conduction through the film. In practical condensers the Nusselt model will strictly only apply at low liquid and vapour rates, and where the flowing condensate film is undisturbed. Turbulence can be induced in the liquid film at high liquid rates, and by shear at high vapour rates. This will generally increase the rate of heat transfer over that predicted using the Nusselt model. The effect of vapour shear and film turbulence are discussed in Volume 1, Chapter 9, see also Butterworth (1978) and Taborek (1974). 

Two flow models are used to estimate the mean condensation coefficient in horizontal tubes stratified flow, Figure 12.45a, and annular flow, Figure 12.45. The stratified flow model represents the limiting condition at low condensate and vapour rates, and the annular model the condition at high vapour and low condensate rates. For the stratified flow model, the condensate film coefficient can be estimated from the Nusselt equation, applying a suitable correction for the reduction in the coefficient caused by... 

For condensation inside horizontal tubes, the Nusselt Equation can be applied with a correction for the reduction in condensing coefficient caused by the accumulation of condensation. The correction usually applied is 0.8. No correction for the number of tubes is required. Thus, for condensation inside horizontal tubes ... 

The Nusselt Equations apply to laminar flow of the condensing film. For horizontal condensation the equations... 

For condensation on a vertical surface, the Nusselt Equation takes the form1 ... 

For vertical condensation, the Nusselt Equations are valid for a laminar film according to ... 

Labuntsov (L2), 1957 Heat transfer to condensate films on vertical and horizontal surfaces. In laminar region, Nusselt equations are corrected for (a) inertia effects, (b) variation of physical properties with temperature, (c) effects of waves. In turbulent region various universal velocity profiles are used. Results compared with experimental data. 

Related Calculations. For low values of Reynolds number (4T//x), the Nusselt equation can be used to predict condensing heat-transfer coefficients for vertical tubes ... 

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... 

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 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... 

The principal features of the derivation consist of a force balance and a heat balance for an element of vapor volume. These are also the principal features of Nusselt s derivation for film condensation. The forces acting on a tiny volume are those due to buoyancy and viscosity. The equation for the heat balance states that the increase in the flow rate of vapor which occurs along an increment of tube circumference corresponds to the heat flow rate in that region. Inasmuch as Nusselt s derivation is available in numerous books, Bromley s equivalent derivation will not be given here. The final equation is... 

The equation for thickness of a falling laminar film was first presented by Nusselt, who used the result to predict heat-transfer coefficients for condensing vapors. Measurements of film thickness on a vertical surface (cos = 1) show that Eq. (5.77) is approximately correct for x 1000, but the thickness actually varies with about 0.45 power of the Reynolds number, and the layers are thinner than predicted at low Nr and thicker than predicted above = 1000. The deviations may be due to ripples or waves in the films, which are apparent even at quite low Reynolds numbers. 

COEFFICIENTS FOR FILM-TYPE CONDENSATION. The basic equations for the rate of heat transfer in film-type condensation were first derived by Nusselt. " The Nusselt equations are based on the assumption that the vapor and liquid at the outside boundary of the liquid layer are in thermodynamic equilibrium, so that the only resistance to the flow of heat is that offered by the layer of condensate flowing downward in laminar flow under the action of gravity. It is also assumed that the velocity of the liquid at the wall is zero, that the velocity of the liquid at the outside of the film is not influenced by the velocity of the vapor, and that the temperatures of the wall and the vapor are constant. Superheat in the vapor is neglected, the condensate is assumed to leave the tube at the condensing temperature, and the physical properties of the liquid are taken at the mean film temperature. 

PI CAL USE OF NUSSELT EQUATIONS. In the absence of high vapor velocities, experimental data check Eqs. (13.13) and (13.14) well, and these equations can be used as they stand for calculating heat-transfer coefficients for film-type condensation on a single horizontal tube. Also, Eq, (13.14) can be used for film-type condensation on a vertical stack of horizontal tubes, where the condensate falls cumulatively from tube to tube and the total condensate from the entire stack finally drops from the bottom tube. The average coefficient for the stack of tubes is less than that for one tube it is given by the equation ... 

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