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.

Laminar Film Condensation on a Sphere

1973 ◽  
Vol 95 (2) ◽  
pp. 174-178 ◽  
Author(s):  
J. W. Yang
Keyword(s):  
Horizontal Cylinder ◽  
Film Condensation ◽  
Stagnation Region ◽  
Similarity Transformations ◽  
Laminar Film ◽  
First Case ◽  
Laminar Film Condensation ◽  
Layer Analysis ◽  
Prandtl Numbers ◽  
Inertia Forces

A boundary-layer analysis is made for laminar film condensation on a sphere. Similarity transformations are made for two cases. The first case includes both the inertia forces and heat convection; the solutions are valid in the upper stagnation region. The second case excludes the inertia forces; the solutions are valid over the entire surface for high Prandtl numbers. Results of heat-transfer rate, condensation rate, and film thickness are presented. Comparisons with a vertical plate and a horizontal cylinder are discussed.

Two-Phase Heat Transfer in Thermosyphon Evacuated-Tube Solar Collectors

1989 ◽  
Vol 111 (4) ◽  
pp. 292-297 ◽  
Author(s):  
Karen R. Den Braven
Keyword(s):  
Heat Transfer ◽  
Surface Waves ◽  
Film Surface ◽  
Detailed Examination ◽  
Film Condensation ◽  
Condensation Process ◽  
Two Phase ◽  
Number Range ◽  
Laminar Film Condensation ◽  
Evacuated Tube

This work analyzes the heat transfer within a tilted thermosyphon and its use in a heat pipe evacuated-tube solar collector. A detailed examination is made of the laminar film condensation process, including the effects of interfacial shear due to the moving vapor. Effects of film surface waves are later included. Including the shear term in the constitutive equations changes the predicted film thickness in the condenser portion of the device by less than one percent, depending on location along the surface. This change causes only a slight increase in the predicted heat transfer. Accounting for surface waves increases the heat transfer rate 10 percent to 50 percent in the Reynolds number range studied. The condenser results are combined with a simple trough model for the evaporator portion of the thermosyphon to give the effective heat-transfer coefficient for the entire tube. Predicted performances of the condenser, the evaporator, and the entire tube compare favorably with available data.

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 Free Convection From a Nonisothermal Cylinder

1965 ◽  
Vol 87 (2) ◽  
pp. 237-241 ◽  
Author(s):  
J. C. Y. Koh ◽  
J. F. Price
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Free Convection ◽  
Surface Temperature ◽  
Vertical Plate ◽  
Numerical Solutions ◽  
Horizontal Cylinder ◽  
Linear Combinations ◽  
All Solutions ◽  
Wall Surface Temperature

Laminar free convection from a nonisothermal horizontal cylinder is analyzed. The wall surface temperature is assumed to be varied in the manner of a1(x/R)2 + a2(x/R)4. Special transformations are devised and employed so that the resulting differential equations and boundary conditions are free of the parameters a1 and a2. These differential equations are solved once and for all; solutions to the original equations for any particular values of a1 and a2 may then be read off easily as linear combinations of the numerical solutions given here. It is found that the dependence of heat transfer from a horizontal cylinder on Prandtl number is practically the same as that from a vertical plate. Furthermore, the heat transfer is greatly influenced by the surface temperature variations.

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.

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.

Normal form method in search for periodic solutions of ordinary differential equations. Case of the fourth order

2018 ◽  
pp. 24-34
Author(s):  
V. F. Edneral ◽  
O. D. Timofeevskaya
Keyword(s):  
Normal Form ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Periodic Solutions ◽  
Fourth Order ◽  
Numerical Solutions ◽  
Numerical Calculations ◽  
Computational Time ◽  
Resonant Normal Form ◽  
Normal Form Method

Introduction:The method of resonant normal form is based on reducing a system of nonlinear ordinary differential equations to a simpler form, easier to explore. Moreover, for a number of autonomous nonlinear problems, it is possible to obtain explicit formulas which approximate numerical calculations of families of their periodic solutions. Replacing numerical calculations with their precalculated formulas leads to significant savings in computational time. Similar calculations were made earlier, but their accuracy was insufficient, and their complexity was very high.Purpose:Application of the resonant normal form method and a software package developed for these purposes to fourth-order systems in order to increase the calculation speed.Results:It has been shown that with the help of a single algorithm it is possible to study equations of high orders (4th and higher). Comparing the tabulation of the obtained formulas with the numerical solutions of the corresponding equations shows good quantitative agreement. Moreover, the speed of calculation by prepared approximating formulas is orders of magnitude greater than the numerical calculation speed. The obtained approximations can also be successfully applied to unstable solutions. For example, in the Henon — Heyles system, periodic solutions are surrounded by chaotic solutions and, when numerically integrated, the algorithms are often unstable on them.Practical relevance:The developed approach can be used in the simulation of physical and biological systems.

scholarly journals Parametric analysis of the heat transfer behavior of the nano-particle ionic-liquid flow between concentric cylinders

2021 ◽  
Vol 13 (6) ◽  
pp. 168781402110240
Author(s):  
Rehan Ali Shah ◽  
Hidayat Ullah ◽  
Muhammad Sohail Khan ◽  
Aamir Khan
Keyword(s):  
Heat Transfer ◽  
Ionic Liquid ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Heat Source ◽  
Ordinary Differential Equations ◽  
Heat Transfer Rate ◽  
Transfer Rate ◽  
Volume Fraction ◽  
Concentric Cylinders

This paper investigates the enhanced viscous behavior and heat transfer phenomenon of an unsteady two di-mensional, incompressible ionic-nano-liquid squeezing flow between two infinite parallel concentric cylinders. To analyze heat transfer ability, three different type nanoparticles such as Copper, Aluminum [Formula: see text], and Titanium oxide [Formula: see text] of volume fraction ranging from 0.1 to 0.7 nm, are added to the ionic liquid in turns. The Brinkman model of viscosity and Maxwell-Garnets model of thermal conductivity for nano particles are adopted. Further, Heat source [Formula: see text], is applied between the concentric cylinders. The physical phenomenon is transformed into a system of partial differential equations by modified Navier-Stokes equation, Poisson equation, Nernst-Plank equation, and energy equation. The system of nonlinear partial differential equations, is converted to a system of coupled ordinary differential equations by opting suitable transformations. Solution of the system of coupled ordinary differential equations is carried out by parametric continuation (PC) and BVP4c matlab based numerical methods. Effects of squeeze number ( S), volume fraction [Formula: see text], Prandtle number (Pr), Schmidt number [Formula: see text], and heat source [Formula: see text] on nano-ionicliquid flow, ions concentration distribution, heat transfer rate and other physical quantities of interest are tabulated, graphed, and discussed. It is found that [Formula: see text] and Cu as nanosolid, show almost the same enhancement in heat transfer rate for Pr = 0.2, 0.4, 0.6.

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 non-alignment stagnation-point flow and heat transfer over a stretching/shrinking surface in a nanofluid

2016 ◽  
Vol 26 (6) ◽  
pp. 1747-1767 ◽  
Author(s):  
Ioan Pop ◽  
Kohi Naganthran ◽  
Roslinda Nazar
Keyword(s):  
Heat Transfer ◽  
Differential Equations ◽  
Stagnation Point ◽  
Numerical Solutions ◽  
Dual Solutions ◽  
Stagnation Point Flow ◽  
Content Type ◽  
Boundary Layer Equations ◽  
Flow And Heat Transfer ◽  
Shrinking Surface

Purpose – The purpose of this paper is to analyse numerically the steady stagnation-point flow of a viscous and incompressible fluid over continuously non-aligned stretching or shrinking surface in its own plane in a water-based nanofluid which contains three different types of nanoparticles, namely, Cu, Al2O3 and TiO2. Design/methodology/approach – Similarity transformation is used to convert the system of boundary layer equations which are in the form of partial differential equations into a system of ordinary differential equations. The system of similarity governing equations is then reduced to a system of first-order differential equations and solved numerically using the bvp4c function in Matlab software. Findings – Unique solution exists when the surface is stretched and dual solutions exist as the surface shrunk. For the dual solutions, stability analysis has revealed that the first solution (upper branch) is stable and physically realizable, while the second solution (lower branch) is unstable. The effect of non-alignment is huge for the shrinking surface which is in contrast with the stretching surface. Practical implications – The results obtained can be used to explain the characteristics and applications of nanofluids, which are widely used as coolants, lubricants, heat exchangers and micro-channel heat sinks. This problem also applies to some situations such as materials which are manufactured by extrusion, production of glass-fibre and shrinking balloon. In this kind of circumstance, the rate of cooling and the stretching/shrinking process play an important role in moulding the final product according to preferable features. Originality/value – The present results are original and new for the study of fluid flow and heat transfer over a stretching/shrinking surface for the problem considered by Wang (2008) in a viscous fluid and extends to nanofluid by using the Tiwari and Das (2007) model.

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