An Analytical Study of Laminar Film Condensation: Part 1—Flat Plates

1961 ◽  
Vol 83 (1) ◽  
pp. 48-54 ◽  
Author(s):  
Michael Ming Chen
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Velocity Gradient ◽  
Film Condensation ◽  
Transfer Coefficients ◽  
Flat Plates ◽  
Integral Boundary ◽  
Boundary Layer Equations ◽  
Laminar Film ◽  
Laminar Film Condensation

The boundary-layer equations of momentum and energy are written in a modified integral form and solved for the case of laminar film condensation along a vertical flat plate. The analysis differs from previous works by employing the more realistic boundary condition of stationary vapor at large distances instead of zero velocity gradient at the interface. Solutions for both the liquid film and vapor boundary layer are given for the case μvρv ≪ μρ. Velocity and temperature profiles are obtained using perturbation method and the modified integral boundary-layer equations. The results show a significant negative velocity gradient at the interface as a result of vapor drag except for small values of kΔt/μλ. Theoretical heat-transfer coefficients are computed and found to be lower than previous theories, especially for low Prandtl numbers. Comparison with experimental heat-transfer data is given. The heat-transfer results are also presented in the form of an approximate formula for ease of application.

An Analytical Study of Laminar Film Condensation: Part 2—Single and Multiple Horizontal Tubes

1961 ◽  
Vol 83 (1) ◽  
pp. 55-60 ◽  
Author(s):  
Michael Ming Chen
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Horizontal Tube ◽  
Vertical Tube ◽  
Film Condensation ◽  
Transfer Coefficients ◽  
Inertia Effects ◽  
Horizontal Tubes ◽  
Laminar Film ◽  
Laminar Film Condensation

The boundary-layer equations for laminar film condensation are solved for (a) a single horizontal tube, and (b) a vertical bank of horizontal tubes. For the single-tube case, the inertia effects are included and the vapor is assumed to be stationary outside the vapor boundary layer. Velocity and temperature profiles are obtained for the case μvρv/μρ ≪ 1 and similarity is found to exist exactly near the top stagnation point, and approximately for the most part of the tube. Heat-transfer results computed with these similar profiles are presented and discussed. For the multiple-tube case, the analysis includes the effect of condensation between tubes, which is shown to be partly responsible for the high observed heat-transfer rate for vertical tube banks. The inertia effects are neglected due to the insufficiency of boundary-layer theory in this case. Heat-transfer coefficients are presented and compared with experiments. The theoretical results for both cases are also presented in approximate formulas for ease of application.

Revisit of Laminar Film Condensation Boundary Layer Theory for Solution of Mixed Convection Condensation With or Without Noncondensables

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
Y. Liao
Keyword(s):  
Boundary Layer ◽  
Mixed Convection ◽  
Forced Convection ◽  
Vapor Condensation ◽  
Boundary Layer Theory ◽  
Film Condensation ◽  
Two Phase ◽  
Boundary Layer Equations ◽  
Laminar Film ◽  
Laminar Film Condensation

This work presents a unique and unified formulation to solve the laminar film condensation two-phase boundary layer equations for the free, mixed, and forced convection regimes in the absence or presence of noncondensables. This solution explores the vast space of mixed convection across the four cornerstones of laminar film condensation boundary layer theory, two established by Koh for pure vapor condensation in the free or forced convection regimes and the other two established by Sparrow corresponding to condensation with noncondensables. This formulation solves the space of mixed convection completely with Koh and Sparrow’s solutions shown to be merely four specific cases of the current solution.

A Boundary-Layer Treatment of Laminar-Film Condensation

1959 ◽  
Vol 81 (1) ◽  
pp. 13-18 ◽  
Author(s):  
E. M. Sparrow ◽  
J. L. Gregg
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Differential Equations ◽  
Prandtl Number ◽  
Vertical Plate ◽  
Film Condensation ◽  
Laminar Film ◽  
Laminar Film Condensation ◽  
Prandtl Numbers ◽  
Mathematical Techniques

The problem of laminar-film condensation on a vertical plate is attacked using the mathematical techniques of boundary-layer theory. Starting with the boundary-layer (partial differential) equations, a similarity transformation is found which reduces them to ordinary differential equations. Energy-convection and fluid-acceleration terms are fully accounted for. Solutions are obtained for values of the parameter cpΔT/hfg between 0 and 2 for Prandtl numbers between 1 and 100. These solutions take their place in the boundary-layer family along with those of Blasius, Pohlhausen, Schmidt and Beckmann, and so on. Heat-transfer results are presented. It is found that the Prandtl-number effect, which arises from retention of the acceleration terms, is very small for Prandtl numbers greater than 1.0. Low Prandtl number (0.003–0.03) heat-transfer results are given in Appendix 2, and a greater effect of the acceleration terms is displayed.

An Integral Treatment of Two-Phase Boundary Layer in Film Condensation

1961 ◽  
Vol 83 (3) ◽  
pp. 359-362 ◽  
Author(s):  
J. C. Y. Koh
Keyword(s):  
Boundary Layer ◽  
Phase Boundary ◽  
Film Condensation ◽  
Two Phase ◽  
Boundary Layer Problem ◽  
Boundary Layer Equations ◽  
Laminar Film ◽  
Laminar Film Condensation ◽  
Integral Treatment ◽  
Two Parameters

By using the integral method, the task of solving the complicated two-phase boundary-layer differential equations in laminar-film condensation has been reduced to the simple work of solving an algebraic equation. It was shown analytically that the parameter [(ρμ)L/(ρμ)v]1/2 can be removed from the film-condensation problem and hence only two parameters, cpΔT/hfg and Pr, are involved. The calculated results in heat transfer and condensate flow rates agree very well with the results from the exact solution of the boundary-layer equations. With [(ρμ)v/(ρμ)L]1/2 remaining as a parameter, it is believed that the present method can be used to solve the analogous two-phase boundary-layer problem in laminar film boiling.

Some Reflections On Sparrow's Work On Similarity Solutions for Laminar Film Condensation On a Vertical Plate

2021 ◽  
Author(s):  
Vijay K. Dhir
Keyword(s):  
Heat Transfer ◽  
Vertical Plate ◽  
Generalized Solution ◽  
Similarity Solutions ◽  
Film Condensation ◽  
Gravitational Acceleration ◽  
Curved Surfaces ◽  
Laminar Film ◽  
Laminar Film Condensation ◽  
Variable Gravity

Abstract In this contribution in honor of Late Prof. E. M. Sparrow, we reflect on the pioneering work of Sparrow and Gregg on the development of similarity solutions for laminar film condensation on a vertical plate. Dhir and Lienhard using this work as a basis developed a generalized solution for isothermal curved surfaces on which gravitational acceleration varied along the surface and for variable gravity situations. Subsequently non-isothermal surfaces were also considered. These studies were publisher earlier in the J. Heat Transfer.

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