Effect of inclination on laminar film condensation

The effect of inclination on laminar film condensation over and under isothermal flat plates is investigated analytically. The complete set of Navier Stokes equations in two dimensions is considered. Analysed as a perturbation problem, the zero-order perturbation represents the boundary layer equations. First and second order perturbations are solved to bring about the leading edge effects. Corresponding velocity and temperature profiles are presented. The results show decrease in heat transfer with larger ∥inclinations∥ from the vertical. Comparison with experimental data of Gerstmann and Griffith indicates a closer agreement of the present results than the analytical results of the same authors. This work was done while the author was at the University of British Columbia, Vancouver, Canada.     

Pure steam condensation model with laminar film in a vertical tube

A new physical model for calculating the liquid film thickness and condensation heat transfer coefficient in a vertical condenser tube is proposed by considering the effects of gravity, liquid viscosity, and vapor flow in the core region of the flow. To estimate the velocity profile in the liquid film, the liquid film was assumed to be in Couette flow forced by the interfacial velocity at the liquid–vapor interface. For simplifying the calculation procedures, the interfacial velocity was estimated by introducing an empirical power-law velocity profile. The resulting film thickness and heat transfer coefficient from the model were compared with the experimental data and the results obtained from the other condensation models. The results demonstrated that the proposed model described the liquid film thinning effect by the vapor shear flow and predicted the condensation heat transfer coefficient from experiments reasonably well.     

An integral procedure for a two-phase boundary layer in laminar film condensation

A general integral procedure has been developed for the analysis of laminar film condensation heat transfer. A solution of the gravity force condensation problem with constant properties is presented, including the effect of the drag due to an initially stationary body of pure saturated vapor. The present integral method, based on the two-phase boundary layer theory, assumes a finite density-viscosity ratio between the liquid and vapor phases, and can, in fact, be employed under almost all possible two dimensional and axisymmetric geometrical configurations. An extensive comparison of the calculation results with available exact solutions reveals an excellent performance of the present integral procedure.     

Theoretical analysis of laminar film condensation in a rotating cylinder with a scraper

The rimming film condensation on the inside wall of a rotating cylinder with a scraper is analyzed. The whole cylinder is divided into two regions, one is the so-called boundary layer region where the radial velocity of the condensate is much smaller than the peripheric velocity so that the boundary layer theory is assumed to be valid; the other is the scraper region where because of the disturbance of the scraper the boundary layer theory does not apply. The boundary layer integral method in the boundary layer region coupling with the integral momentum theorem across the scraper region provides a method to determine the velocity, temperature, and film thickness distributions, and heat transfer coefficients. An extensive discussion about the previous models is given. The sublayer flow rate constancy principle and the variability principle of the boundary layer thickness (therefore the interface velocity) at the scraper position with respect to the rotational speed are proposed. The present model greatly improved the prediction of the average heat transfer coefficient. Received on 5 January 1998     

Local nonsimilarity method for the two-phase boundary layer in mixed convection laminar film condensation

The two-phase boundary layer in laminar film condensation was solved by Koh for the free convection regime and forced convection regime using the similarity method. But the problem on mixed convection does not admit similarity solutions. The current work develops a local nonsimilarity method for the full spectrum of mixed convection, with a generic boundary layer formulation reduced to two specific cases mathematically identical to Koh’s formulations on the two limiting cases for either free or forced convection. Other solution methods for mixed convection in the literature are compared and critically evaluated to ensure a high level of accuracy in the current method. The current solution is used to extend Fujii’s correlation for mixed convection to the region where the energy convection effect is significant but has been hitherto neglected. The modified Fujii correlation provides a practical engineering tool for evaluating laminar film condensation with a mixed convection boundary layer.     

A two-phase model for laminar film condensation from steam-air mixtures in vertical parallel-plate channels

Detailed results are presented for laminar film condensation from steam-air mixtures flowing downward in vertical flat-plate channels. The mixture flow is laminar and saturation conditions prevail at the inlet. A fully coupled implicit numerical approach is used that achieves excellent convergence behavior, even for high inlet gas mass fractions. The detailed results include velocity, temperature, and gas mass fraction profiles, as well as axial variations of film thickness, pressure gradient and Nusselt number. The effects of a wide range of changes in the four independent variables (the inlet-to-wall temperature difference and the inlet values of gas concentration, Reynolds number, and pressure) on the film thickness, axial pressure gradient, and the local and average Nusselt numbers are carefully examined. It was found that increases in inlet concentration of noncondensable gas caused significant decreases in the film thickness, local Nusselt number, and axial pressure gradient. An analytical solution for the film thickness and velocity field at the end of condensation path was developed and shown to be the asymptotic value of the numerical results for large distances along the channel.     

Heat transfer in laminar film condensation; local film resistance and local enthalpy allowing for variable viscosity and subcooling

In contrast to the conventional method of calculation account was taken of the dependence of viscosity on temperature and of the non-linear temperature profile due to subcooling of the condensate. Fully developed velocity and temperature profiles and a constant heat flux along the flow path are assumed. The local thickness of the film can be calculated with the usual equations provided the reference temperature of Drew [3] and Gregorig [4] for the viscosity (three-quarters of the wall temperature plus one-quarter of the film surface temperature) is used. The calculation of the local heat transfer coefficient and the mean condensate temperature (adiabatic mixing temperature), however, requires special equations taking into account the influences mentioned above.     

Forced convection laminar film condensation on an inclined elliptical tube in the absence of gravity

In this paper, the problem of forced convection dominated laminar film condensation, in the absence of gravity, on an inclined elliptical tube is investigated theoretically. In this analysis, the interfacial vapour shear stress is modelled following Shekriladze approach. By employing the method of characteristics, expressions are analytically derived for calculating the local film thickness as well as the local and mean Nusselt numbers. The results show that the mean Nusselt number, enhances with the increase in the tube ellipticity. For the practical ellipticity range: 0.8 e 0.92, this enhancement is found from 7 to 14% compared to a circular tube with the same length and equivalent condensation surface area.

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Stability of a laminar film flow

The problem of the wave motion of a liquid layer was first investigated by Kapitsa [1, 2], who gave an approximate analysis of the free flow and flow in contact with gas stream, and evaluated the influence of the heat transfer processes on the flow. The problem of the stability of such a flow was studied in detail by Benjamin [3] and Yih [4, 5], These authors proposed seeking the solution of the resulting Orr-Sommerfeld equation in the form of a series in a small parameter and developed a corresponding method of successive approximations. As the small parameter [3–5], they made use of the product of the disturbance wave number and the Reynolds number. In these studies, the tangential stress on the free surface was taken equal to zero, and the fluid film was always considered essentially plane. At the same time, there are certain types of problems of considerable interest in which neither of these assumptions is satisfied. A good example might be the problem on the stability of the annular regime of two-phase flow in pipes and capillaries, when the basic stream of one fluid is separated from the pipe walls by an annular layer of another fluid. In this case, the interface has a finite radius of curvature and the tangential stress on the interface may be significantly different from zero.In the present paper, the problem of the flow stability of a fluid layer with respect to small disturbances of the boundary surface is considered with account for both the finite radius of curvature of the boundary surface and the nonzero hydrodynamic friction at the boundary. The film is assumed to be quite thin. This enables us, firstly, to consider the Reynolds number small, to use the general method of [5], and, second ly, to consider the film thickness sufficiently small in comparison with the radius of curvature of the substrate on which the film lies. Furthermore, for evaluating the stability of the laminar flow of the curved film we can use the results obtained for a plane film with account for the terms which depend on the curvature of the substrate.As a rule, previous studies have considered only one-dimensional disturbances of the boundary surface. In the present paper, in the first approximation, the stability is examined in relation to two-dimensional disturbances of this surface, corresponding to three-dimensional flow disturbances.As an example, the results obtained are applied to the investigation of the stability of the free flow of a layer of fluid over an inclined plane under the sole influence of gravity.     

Forced convective film condensation inside vertical tubes

A theoretical study of forced convective film condensation inside vertical tubes is presented. We propose a unified procedure for predicting the pressure gradient and condensation heat transfer coefficient of a vapor flowing turbulently in the core and associated with laminar or turbulent film on the tube wall. The analysis for the vapor flows is performed under the condition that the velocity profiles are locally self-similar. The laminar and turbulent film models equate the gravity, pressure and viscous forces, and consider the effect of interfacial shear. The transition from laminar to turbulent film depends not only on the liquid Reynolds number but also on the interfacial shear stress. In this work we also proposed a new eddy viscosity model which is divided into three regions: the inner region in liquid condensate near the wall; the interface region including both liquid and vapor; and the outer region for the vapor core. Comparisons of the theory with some published experimental data showed good agreement.     

Laminar film condensation on a horizontal disk

Heat and Mass Transfer - The problem of two-dimensional, steady film condensation on a finite-size, isothermal horizontal disk is studied for the case in which a cold plate faces upwards. The...     

Two phase boundary layers in laminar film boiling

The laminar film boiling is analyzed by means of an integral procedure. The method treats the film boiling as a two phase boundary layer problem; thereby the effect of the interfacial shear on the heat transfer rate can be investigated. The problem is attacked by simultaneously solving the vapor and liquid boundary layer equations. An extensive comparison of the predicted results with the exact solutions substantiates the validity of the present integral procedure. Even the details of the velocity and temperature profiles turn out to be in close agreement with the exact solutions.     

Study on laminar film condensation of saturated steam on a vertical flat plate for consideration of various physical factors including variable thermophysical properties

The extended theory of the steady state laminar film condensation process of pure saturated vapour at atmospheric pressure on an isothermal vertical flat plate is established. Its equations provide a complete account of the physical process for consideration of various physical factors including variable thermophysical properties, except for surface tension at the liquid-vapour film interface. First, similarity considerations are proposed to transform the governing system of partial differential equations and its boundary conditions into the corresponding dimensionless system. Then, the dimensionless new system is computed numerically in two steps: First neglecting shear force at the interface, so that the initial values of the boundary conditionsW _{xl, s} andW _{yl, s} are obtained. Then, the calculations of a problem of the three-point boundary-value for coupling the equations of liquid film with those of vapour film are carried out. Furthermore, the correlations for heat transfer coefficient and mass flow rate are proposed by analysis of heat and mass transfer and it is found that the heat transfer coefficient is function of dimensionless temperature gradient $\dot L$ , and that the condensate mass flow rate is function of the mass flow rate parameter (η _lδ W _{xl, s} ? 4W _{yl, s} )of liquid. In addition, the corresponding heat and mass transfer correlations expressed by subcooled temperature Δt are developed. According to Nusselt's theory four different assumptions are set up for an investigation of the effects of the film condensation of saturated vapour, so that the validity of Nusselt's theory can be further clarified. Quantitative comparisons from the results of the heat transfer coefficient and mass flow rate of the condensate indicate that the effect of variable thermophysical properties on the heat and mass transfer is appreciable. The effect of thermal convection in the condensate film is obviously larger than those of shear force at liquid-vapour interface, and the effect of the inertia in the condensate film is very small. Finally, it is also shown that Nusselt's theory, in using Drew reference temperature, will decrease the heat transfer coefficient by at most 5.11%, and will increase the mass flow rate of the condensate by at most 2.45%, provided that the effect of the surface tension is not taken into account.     

Turbulent film condensation on an isothermal cone surface

It is an investigation of turbulent film condensation on an isothermal cone. The present paper describes the eddy diffusivity of two turbulent models. And then it discusses the film thickness and heat transfer characteristics under the different turbulent models. The results show the mean heat transfer coefficient on two forms of eddy diffusivity, and there is a variation on the two models. Furthermore, the current results are compared with those generated by previous theoretical investigations. It is found that in high vapor velocity, the mean heat transfer was greater than that of the laminar flow theory. Under the high vapor velocity region, the eddy effect will be an important factor for the heat transfer of turbulent condensate film. Besides, in the low vapor velocity region, the eddy diffusivity seldom influences the heat transfer of condensate film.     

Turbulent film heat-transfer coefficients during condensation in tubes

Equations for the local film heat-transfer coefficient with a turbulent film are obtained based on an approximate boundary layer treatment and compared with experiment and other well-known predictive methods.     

Turbulent film condensation on a horizontal elliptical tube

An investigation has been made of turbulent film condensation on a horizontal elliptical tube. The present study is based on Colburn analogy [1] and potential flow theory to determine the high tangential velocity of vapor flow at the boundary layer and to define the local interfacial shear owing to high velocity vapor flow across the tube surface. The condensate film flow and local/or mean heat transfer characteristics from a horizontal elliptical tube with variable ellipticities, e, under the influence of Froude number, sub-cooling parameter and system pressure have been performed. The present result for dimensionless mean heat transfer coefficient reduces to the same result obtained by Sarma et al.s [2] e=0 (circular tube). Compared with laminar model by Yang and Hsu [3], the present turbulent model shows in better agreement with Michaels experimental data [4] (for e=0). The dependence of mean Nusselt coefficient on the effect of n (power of Reynolds) [1] is also discussed.     

Combined forced and natural convection in laminar film flow

A mathematical model for the flow and heat transfer in a gravity-driven liquid film is presented, in which the strict Boussinesq approximation is adopted to account for buoyancy. A similarity transformation reduces the governing equations to a coupled set of ordinary differential equations. The resulting two-parameter problem is solved numerically for Prandtl numbers ranging from 1 to 1000. Favourable buoyancy arises when the temperatureT _w of the isothermal surface is lower than the temperatureT ₀ of the incoming fluid, and the principal effects of the aiding buoyancy are to increase the wall shear and heat transfer rate. For unfavourable buoyancy (T _w>T ₀), the buoyancy force and gravity act in opposite directions and the flow in the film boundary layer decelerates, whereas the friction and heat transfer are reduced. The observed effects of buoyancy diminish appreciably for higher Prandtl numbers.