A Theory of Rotating Condensation

1959 ◽  
Vol 81 (2) ◽  
pp. 113-119 ◽  
Author(s):  
E. M. Sparrow ◽  
J. L. Gregg
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Large Body ◽  
Rotating Disk ◽  
Disk Surface ◽  
Film Condensation ◽  
Navier Stokes ◽  
Torque Moment ◽  
Prandtl Numbers ◽  
Energy Equations

An analysis is made for film condensation on a rotating disk situated in a large body of pure saturated vapor. The centrifugal field associated with the rotation sweeps the condensate outward along the disk surface, and gravity forces need not be involved. The problem is formulated as an exact solution of the complete Navier-Stokes and energy equations. Numerical solutions are obtained for Prandtl numbers between 0.003 and 100 and for cpΔT/hfg in the range 0.0001 to 1.0. Results are given for the heat transfer, as well as for the condensate layer thickness, torque moment, and temperature and velocity profiles.

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.

Numerical Solutions of Transients in Pneumatic Networks—Transmission-Line Calculations

1968 ◽  
Vol 35 (3) ◽  
pp. 588-595 ◽  
Author(s):  
S. Tsao
Keyword(s):  
Wave Propagation ◽  
Transmission Lines ◽  
Numerical Solutions ◽  
Navier Stokes ◽  
Damping Factor ◽  
Temperature Profiles ◽  
Hypothetical Case ◽  
One Dimensional ◽  
Simplifying Assumptions ◽  
Energy Equations

Equations governing the damped wave propagation along transmission lines are obtained from the Navier-Stokes and energy equations by making certain simplifying assumptions. The flow considered is essentially one-dimensional. However, radial variations of the velocity and temperature profiles must be considered, because the damping factor is directly dependent on them. The equations are integrated by numerical methods. A hypothetical case is computed as an example.

Heat Transfer From a Circular Cylinder at Low Reynolds Numbers

1998 ◽  
Vol 120 (1) ◽  
pp. 72-75 ◽  
Author(s):  
V. N. Kurdyumov ◽  
E. Ferna´ndez
Keyword(s):  
Circular Cylinder ◽  
Energy Equation ◽  
Numerical Solutions ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Order Unity ◽  
Navier Stokes Equations ◽  
Wide Range ◽  
Prandtl Numbers ◽  
High Degree

A correlation formula, Nu = W0(Re)Pr1/3 + W1(Re), that is valid in a wide range of Reynolds and Prandtl numbers has been developed based on the asymptotic expansion for Pr → ∞ for the forced heat convection from a circular cylinder. For large Prandtl numbers, the boundary layer theory for the energy equation is applied and compared with the numerical solutions of the full Navier Stokes equations for the flow field and energy equation. It is shown that the two-terms asymptotic approximation can be used to calculate the Nusselt number even for Prandtl numbers of order unity to a high degree of accuracy. The formulas for coefficients W0 and W1, are provided.

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.

Influence of temperature dependent physical properties on liquid metal droplet impact dynamics

2021 ◽  
pp. 1-15
Author(s):  
Akash Chowdhury ◽  
Anandaroop Bhattacharya ◽  
Partha Bandyopadhyay
Keyword(s):  
Surface Tension ◽  
Numerical Solutions ◽  
Substrate Surface ◽  
Metal Droplet ◽  
Navier Stokes ◽  
Complex Flow ◽  
Liquid Metal Droplet ◽  
Aluminium Copper ◽  
Energy Equations ◽  
Surface Tension Forces

Abstract The dynamics of a metal droplet impacting on a substrate surface has been studied in the paper numerically. Numerical solutions of the Navier-Stokes and Energy equations show the evolution of the droplet as it spreads upon impact with the substrate while simultaneously undergoing solidification. The interplay of the different forces including inertia, viscous and surface tension, coupled with solidification of the molten material in layers lead to complex flow dynamics. The change in density and viscosity owing to change in temperature resulting from the cooling process, is found to influence the spreading of the droplet significantly. The model was exercised for three different materials viz. aluminium, copper and nickel to determine the final splat radius as well as spreading time. The surface tension forces as well as solidification rates were found to be the dominant factors in determining the above parameters as well as the shape of the splat during spreading. The results were found to be in good agreement with existing analytical model.

Laminar Film Condensation on a Vertical Melting Surface

1976 ◽  
Vol 98 (1) ◽  
pp. 108-113 ◽  
Author(s):  
M. Epstein ◽  
D. H. Cho
Keyword(s):  
Liquid Film ◽  
Saturated Vapor ◽  
Previous Treatment ◽  
Film Condensation ◽  
Full Account ◽  
Component System ◽  
Laminar Film ◽  
Laminar Film Condensation ◽  
Melting Surface ◽  
Prandtl Numbers

Laminar film condensation of a saturated vapor on a vertical melting surface is treated theoretically, with emphasis on departures from a previous treatment produced by: (a) arbitrary liquid Prandtl numbers and (b) condensation-melting systems involving two materials of immiscible liquids. An integral method is utilized which takes full account of the effects of both liquid film inertia and shear force at the condensing vapor-liquid film interface. For a one-component system accurate numerical results for the melting rates are displayed graphically and define the range of validity of a simple treatment of this problem based on Nusselt’s method. For a two-component system, illustrative calculations are made for the condensation of a refrigerant vapor on melting ice.

Nonsimilar Boundary-Layer Heat Transfer of a Rotating Cone in Forced Flow

1967 ◽  
Vol 89 (2) ◽  
pp. 139-145 ◽  
Author(s):  
J. C. Y. Koh ◽  
J. F. Price
Keyword(s):  
Heat Transfer ◽  
Boundary Layer ◽  
Numerical Solutions ◽  
Cone Angle ◽  
Rotating Disk ◽  
Forced Flow ◽  
Degenerate Problem ◽  
Prandtl Numbers ◽  
Half Cone ◽  
Half Cone Angle

The nonsimilar boundary-layer flow and heat transfer of a cone rotating in a forced-flow field are investigated. Numerical solutions are shown for a half-cone angle of 53.5 deg with parameters (vw/ue)2 ranging from 0 to 20, and with Prandtl numbers from 0.2 to 10. With a half-cone angle of 90 deg (so that one has a rotating disk), the degenerate problem is solved in the same manner.

The Classical Nusselt Problem of Film Condensation: Direct Steady and Unsteady Computational Simulations That Yield Results on Stability and Wave Effects

2005 ◽  
Author(s):  
L. Phan ◽  
S. L. Post ◽  
A. Narain
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Vertical Wall ◽  
Flow Instabilities ◽  
Film Condensation ◽  
Linear Wave ◽  
Governing Equations ◽  
Transfer Rates ◽  
Transverse Vibrations ◽  
Wave Effects

Accurate steady and unsteady numerical solutions of the full 2D governing equations for the Nusselt problem (film condensation of quiescent saturated vapor on a vertical wall) are presented and related to known results. The problem, solved accurately up to film Reynolds number of 60 (Reδ ≤ 60), establishes various features of the well known steady solution and reveals the interesting phenomena of stability, instability and non-linear wave effects. The wave effects are shown to arise from the intrinsic flow instabilities as well as sensitivity to ever present minuscule transverse vibrations of the condensing surface. The results also suggest ways to enhance wave fluctuations and heat transfer rates.

Nonlinear Stability of the Classical Nusselt Problem of Film Condensation and Wave Effects

2006 ◽  
Vol 74 (2) ◽  
pp. 279-290 ◽  
Author(s):  
L. Phan ◽  
A. Narain
Keyword(s):  
Numerical Solutions ◽  
Saturated Vapor ◽  
Vertical Wall ◽  
Film Condensation ◽  
Governing Equations ◽  
Transfer Rates ◽  
Laminar To Turbulent Transition ◽  
Turbulent Transition ◽  
Transverse Vibrations ◽  
Wave Effects

Accurate steady and unsteady numerical solutions of the full two-dimensional (2D) governing equations for the Nusselt problem (film condensation of quiescent saturated vapor on a vertical wall) are presented and related to known results. The problem, solved accurately up to film Reynolds number of 60 (Reδ⩽60), establishes various features of the well-known steady solution and reveals the interesting phenomena of stability, instability, and nonlinear wave effects. It is shown that intrinsic flow instabilities cause the wave effects to grow over the well-known experiments-based range of Reδ⩾30. The wave effects due to film flow’s sensitivity to ever-present minuscule transverse vibrations of the condensing surface are also described. The results suggest some ways of choosing wall noise—through suitable actuators—that can enhance or dampen wave fluctuations and thus increase or decrease heat transfer rates over the laminar-to-turbulent transition zone.

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