Film Condensation Over a Horizontal Cylinder for Combined Gravity and Forced Flow

1985 ◽  
Vol 107 (3) ◽  
pp. 687-695 ◽  
Author(s):  
M. di Marzo ◽  
M. J. Casarella
Keyword(s):  
Froude Number ◽  
Numerical Solutions ◽  
Driving Forces ◽  
Horizontal Cylinder ◽  
Forced Flow ◽  
Film Condensation ◽  
Vapor Flow ◽  
Governing Equations ◽  
Laminar Film Condensation ◽  
Wide Range

The problem of laminar film condensation of a saturated vapor flowing over a cold horizontal cylinder is investigated. A rigorous formulation of the governing equations for the vapor boundary layer and the condensed liquid film, including both the gravity-driven body forces and the imposed pressure gradient caused by the vapor flow, is presented. A generalized transformation of the governing equations allows a wide range of Froude numbers to be investigated. A unique value of the Froude number is defined which allows a distinction between the gravity-dominated flow (Fr→0) and the forced flow (Fr→∞) and basically defines the overlap region for the two solution domains. Numerical solutions are obtained in the merging flow regions controlled by both driving forces. The effects of density/viscosity ratio at the liquid-vapor interface, Prandtl number, Jakob number, and Froude number on the heat transfer characteristics are presented.

Forced Flow of Vapor Condensing Over a Horizontal Plate (Problem of Cess and Koh): Steady and Unsteady Solutions of the Full 2D Problem

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
S. Kulkarni ◽  
A. Narain ◽  
S. Mitra
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Threshold Value ◽  
Steady Solution ◽  
Forced Flow ◽  
Film Condensation ◽  
Horizontal Plate ◽  
Governing Equations ◽  
Long Time ◽  
Steady Solutions

Accurate steady and unsteady numerical solutions of the full 2D governing equations—which model the forced film condensation flow of saturated vapor over a semi-infinite horizontal plate (the problem of Cess and Koh)—are obtained over a range of flow parameters. The results presented here are used to better understand the limitations of the well-known similarity solutions given by Koh. It is found that steady/quasisteady filmwise solution exists only if the inlet speed is above a certain threshold value. Above this threshold speed, steady/quasisteady film condensation solutions exist and their film thickness variations are approximately the same as the similarity solution given by Koh. However, these steady solutions differ from the Koh solution regarding pressure variations and associated effects in the leading part of the plate. Besides results based on the solutions of the full steady governing equations, this paper also presents unsteady solutions that characterize the steady solutions’ attainability, stability (response to initial disturbances), and their response to ever-present minuscule noise on the condensing-surface. For this shear-driven flow, the paper finds that if the uniform vapor speed is above a threshold value, an unsteady solution that begins with any reasonable initial-guess is attracted in time to a steady solution. This long time limiting solution is the same—within computational errors—as the solution of the steady problem. The reported unsteady solutions that yield the steady solution in the long time limit also yield “attraction rates” for nonlinear stability analysis of the steady solutions. The attraction rates are found to diminish gradually with increasing distance from the leading edge and with decreasing inlet vapor speed. These steady solutions are generally found to be stable to initial disturbances on the interface as well as in any flow variable in the interior of the flow domain. The results for low vapor speeds below the threshold value indicate that the unsteady solutions exhibit nonexistence of any steady limit of filmwise flow in the aft portion of the solution. Even when a steady solution exists, the flow attainability is also shown to be difficult (because of waviness and other sensitivities) at large downstream distances.

Laminar Film Condensation of a Flowing Vapor on a Horizontal Cylinder at Normal Gravity

1969 ◽  
Vol 91 (4) ◽  
pp. 495-501 ◽  
Author(s):  
V. E. Denny ◽  
A. F. Mills
Keyword(s):  
Boundary Layer ◽  
Normal Gravity ◽  
Numerical Solutions ◽  
Horizontal Cylinder ◽  
Vertical Surface ◽  
Film Condensation ◽  
Laminar Film ◽  
Laminar Film Condensation ◽  
Fluid Properties ◽  
Finite Difference Analog

An analytical solution, based on the Nusselt assumptions, has been obtained for laminar film condensation of a flowing vapor on a horizontal cylinder. In so doing, a reference temperature for evaluating locally variable fluid properties is defined in the form Tr = Tw + α (Ts − Tw) and accounts for both the effects of fluid property variations and minor errors introduced by the assumptions in the analysis. Verification was obtained by comparison with exact numerical solutions based on a finite-difference analog to the conservation equations in boundary-layer form. In the analytical as well as the numerical developments, vapor drag was accounted for through an asymptotic solution of the vapor boundary layer under strong suction. It was found that, for angles up to 140 deg, there was less than a 2 percent discrepancy between the analytical predictions and the numerical results. As 180 deg is approached an increased discrepancy is expected due to a gross violation of the Nusselt assumptions. The values of the reference parameter α, which were previously derived for condensation on a vertical surface, were found to be appropriate for the horizontal cylinder as well.

Laminar Condensation Heat Transfer on a Horizontal Cylinder

1959 ◽  
Vol 81 (4) ◽  
pp. 291-295 ◽  
Author(s):  
E. M. Sparrow ◽  
J. L. Gregg
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Horizontal Cylinder ◽  
Film Condensation ◽  
Number Range ◽  
Laminar Film Condensation ◽  
Layer Analysis ◽  
Inertia Forces

A boundary-layer analysis is made for laminar film condensation on a horizontal cylinder. The formulation includes both the inertia forces and energy convection terms, which are neglected in Nusselt’s simple theory. A similarity transformation, valid over most of the cylinder, is found which reduces the partial differential equations of the problem (the conservation laws) to ordinary differential equations. Numerical solutions of the resulting ordinary differential equations are available for the Prandtl number range from 0.003 to 100. Heat-transfer results are presented and discussed.

Similar Solutions for Laminar Film Condensation with Adverse Pressure Gradients

1973 ◽  
Vol 95 (2) ◽  
pp. 268-270 ◽  
Author(s):  
P. M. Beckett
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Film Condensation ◽  
Pressure Gradients ◽  
Two Dimensional ◽  
Laminar Film ◽  
Approximate Models ◽  
Laminar Film Condensation ◽  
Similar Solutions

Steady two-dimensional laminar film condensation is investigated when the saturated vapor has the Falkner–Skan mainstream. Numerical solutions and approximate models are discussed with reference to other published work.

Fully Coupled Elliptic Numerical Model for Film Condensation From Vapour-Gas Mixtures in Vertical Parallel Plate Channels

2014 ◽  
Author(s):  
Foad Hassaninejadfarahani ◽  
Scott Ormiston
Keyword(s):  
Mass Fraction ◽  
Parallel Plate ◽  
Thermal Power ◽  
Industrial Applications ◽  
Sharp Interface ◽  
Film Condensation ◽  
Governing Equations ◽  
Laminar Film ◽  
Laminar Film Condensation ◽  
Fully Coupled

Laminar film condensation is an important phenomenon which occurs in numerous industrial applications such as refrigeration, chemical processing, and thermal power generation industries. It is well known that film condensation heat transfer is greatly reduced in the presence of a non-condensing gas. The present work performs a numerical analysis of the steady-state, laminar film condensation from a vapour-gas mixture in vertical parallel plate channels to demonstrate a computer model that could assist engineering analysts designing systems involving these phenomena. The present model has three new aspects relative to other current work. First, the complete elliptic two-dimensional governing equations are solved in both phases. Thus, the entire channel domain is solved rather than using an approach that marches along the channel from inlet to a prescribed length. Second, a dynamically determined sharp interface is used between the phases. This sharp interface is determined during the solution on a non-orthogonal structured mesh. Third, the governing equations are solved in a fully-coupled approach. The equations for two velocities, pressure, temperature, and gas mass fraction are solved in a coupled method simultaneously for both phases. Discretisation has been done based on a finite volume method and a co-located variable storage scheme. An in-house computer code was developed to implement the numerical solution scheme. Detailed results are presented for laminar film condensation from steam-air mixtures flowing in vertical parallel-plate channels. The results include velocity and pressure profiles, as well as axial variations of film thickness, Nusselt number and interface gas mass fraction. Detailed comparisons are made with results from a parabolic solution approach.

Laminar Film Condensation From a Steam-Air Mixture Undergoing Forced Flow Down a Vertical Surface

1971 ◽  
Vol 93 (3) ◽  
pp. 297-304 ◽  
Author(s):  
V. E. Denny ◽  
A. F. Mills ◽  
V. J. Jusionis
Keyword(s):  
Liquid Film ◽  
Numerical Solution ◽  
Finite Difference Methods ◽  
Vertical Surface ◽  
Forced Flow ◽  
Film Condensation ◽  
Two Phase ◽  
Noncondensable Gas ◽  
Laminar Film ◽  
Laminar Film Condensation

An analytical study of the effects of noncondensable gas on laminar film condensation of vapor under going forced flow along a vertical surface is presented. Due to the markedly nonsimilar character of the coupled two-phase-flow problem, the set of parabolic equations governing conservation of momentum, species, and energy in the vapor phase was solved by means of finite-difference methods using a forward marching technique. Interfacial boundary conditions for the numerical solution were extracted from a locally valid Nusselt-type analysis of the liquid-film behavior. Locally variable properties in the liquid were treated by means of the reference-temperature concept, while those in the vapor were treated exactly. Closure of the numerical solution at each step was effected by satisfying overall mass and energy balances on the liquid film. A general computer program for solving the problem has been developed and is applied here to condensation from water-vapor–air mixtures. Heat-transfer results, in the form q/qNu versus x, are reported for vapor velocities in the range 0.1 to 10.0 fps with the mass fraction of air ranging from 0.001 to 0.1. The temperature in the free stream is in the range 100–212 deg F, with overall temperature differences ranging from 5 to 40 deg F. The influence of noncondensable gas is most marked for low vapor velocities and large gas concentrations. The nonsimilar character of the problem is especially evident near x = 0, where the connective behavior of the vapor boundary layer is highly position-dependent.

scholarly journals Mixed-convection laminar film condensation on a semi-infinite vertical plate

1995 ◽  
Vol 300 ◽  
pp. 207-229 ◽  
Author(s):  
Jian-Jun Shu ◽  
Graham Wilks
Keyword(s):  
Vertical Plate ◽  
Body Force ◽  
Numerical Solutions ◽  
Force Convection ◽  
Film Condensation ◽  
Approximate Methods ◽  
Saturated Vapour ◽  
Laminar Film ◽  
Laminar Film Condensation ◽  
Infinite Vertical Plate

The flow of a uniform stream of pure saturated vapour past a cold, semi-infinite vertical plate is examined. The formulation incorporates the limits of both pure forcedconvection and pure body-force-convection laminar film condensation. Detailed asymptotic and exact numerical solutions are obtained and comparisons drawn with approximate methods and experimental results reported in the literature.

Mixed-Convection Laminar Film Condensation on an Inclined Elliptical Tube

2000 ◽  
Vol 123 (2) ◽  
pp. 294-300 ◽  
Author(s):  
M. Mosaad
Keyword(s):  
Heat Transfer ◽  
Mixed Convection ◽  
Circular Tube ◽  
Film Condensation ◽  
Equivalent Diameter ◽  
Laminar Film ◽  
Laminar Film Condensation ◽  
Wide Range ◽  
Convection Dominated ◽  
The Heat Transfer Coefficient

The present theoretical study concerns with mixed-convection laminar film condensation outside an inclined elliptical tube with isothermal surface. The assumptions used are as in the classical Nusselt-Rohsenow theory, however, with considering the interfacial vapor shear by extending a circular-tube shear model developed in a previous study. An equivalent diameter, based on equal surface area, is introduced in the analysis to enable comparison with circular tubes. For zero ellipticity, the approach simplifies to the circular tube model developed in our previous work. A numerical solution has been obtained for a wide range of the independent parameters. The results indicate that the heat transfer performance of the inclined elliptical tube enhances with the increase of tube ellipticity compared to an inclined circular tube of equivalent diameter. For forced-convection-dominated film condensation, the rate of this enhancement in the heat transfer coefficient is found smaller than that for pure-free-convection film.

Direct Computational Simulations for Condensation Inside Tubes and Channels

2005 ◽  
Author(s):  
L. Phan ◽  
X. Wang ◽  
S. Kulkarni ◽  
A. Narain
Keyword(s):  
Numerical Solutions ◽  
Vertical Tube ◽  
Film Condensation ◽  
Vapor Flow ◽  
Steady Flows ◽  
Channel Geometry ◽  
Pressure Drops ◽  
Condensing Flows ◽  
Gravity Driven ◽  
Internal Condensing Flows

The paper presents accurate numerical solutions of the full 2D governing equations for steady and unsteady laminar/laminar internal condensing flows of pure vapor (R-113 and FC-72) inside a vertical tube and a channel. The film condensation is on the inside wall of a tube or one of the walls of a channel (the lower wall in case of a downward sloping channel). The new geometry in this paper is the cylindrical in-tube geometry with axisymmetric flows (vertical 1g or 0g flows). The new results encompass both the cylindrical and the earlier studied channel geometry. Exit condition specifications are again found to be important. The computations are able to predict whether or not a steady flow exists under a natural exit condition (selected from a range of choices available at the exit). If natural steady/quasi-steady flows exist — as is shown to be the case for gravity dominated or strong shear dominated condensate flows — the computations are able to predict both the natural exit condition and the associated condensate flow’s point of transition from stable to unstable behavior. Compared to gravity driven, shear driven cases (zero gravity or horizontal cases) tend to destabilize easier and generally have much larger pressure drops, much slower wave speeds, much larger role of surface tension, and much narrower flow regime boundaries within which the vapor flow can be modelled incompressible. It is found that only in gravity driven cases, be it vertical in-tube or inclined channel geometry, interfacial waves are able to cause a concurrent enhancement in heat transfer rates along with an enhancement in interfacial shear. Also it is found that this enhancement is significant if the condensing surface noise is in resonance with the intrinsic waves.

Laminar Film Condensation on a Porous Horizontal Tube With Uniform Suction Velocity

1965 ◽  
Vol 87 (1) ◽  
pp. 95-102 ◽  
Author(s):  
N. A. Frankel ◽  
S. G. Bankoff
Keyword(s):  
Heat Transfer ◽  
Perturbation Technique ◽  
Porous Plate ◽  
Horizontal Tube ◽  
Film Condensation ◽  
Suction Velocity ◽  
Uniform Suction ◽  
Laminar Film Condensation ◽  
Wide Range ◽  
Perturbation Parameters

The analysis of Bankoff and Jain [10] of film condensation on a vertical porous plate with uniform suction velocity is extended to the case of a horizontal porous tube. Integral momentum and energy balances are written for the system, including the effects of interface drag and condensate heat capacity, and the dimensionless equations are solved using a perturbation technique. All dependent variables are expressed in a double power series in the two perturbation parameters, ξ = kΔt/μλ (acceleration parameter) and α (dimensionless suction velocity), and the resulting equations are solved up to the second order perturbation. An asymptotic solution valid for high values of α is derived, and this solution together with the perturbation solution describes the system for a wide range of α. The case of heat transfer in a zero gravity field is treated, and the Nusselt number is found to be directly proportional to the suction velocity. Based on the results it is concluded that significant increases in heat transfer are possible with the use of suction.

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