Analysis of Conduction-Controlled Rewetting of a Vertical Surface

1975 ◽  
Vol 97 (2) ◽  
pp. 161-165 ◽  
Author(s):  
C. L. Tien ◽  
L. S. Yao
Keyword(s):  
Initial Temperature ◽  
Empirical Relation ◽  
Vertical Surface ◽  
Dimensionless Temperature ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
One Dimensional ◽  
Functional Relationships ◽  
Semi Empirical ◽  
Biot Numbers

The present paper presents a two-dimensional analysis of conduction-controlled rewetting of a vertical surface, whose initial temperature is greater than the rewetting temperature. The physical model consists of an infinitely extended vertical slab with the surface of the dry region adiabatic and the surface of the wet region associated with a constant heat transfer coefficient. The physical problem is characterized by three parameters: the Peclet number or the dimensionless wetting velocity, the Biot number, and a dimensionless temperature. Limiting solutions for large and small Peclet numbers obtained by utilizing the Wiener-Hopf technique and the kernel-substitution method exhibit simple functional relationships among the three dimensionless parameters. A semi- empirical relation has been established for the whole range of Peclet numbers. The solution for large Peclet numbers possesses a functional form different from existing approximate two-dimensional solutions, while the solution for small Peclet numbers reduces to existing one-dimensional solution for small Biot numbers. Discussion of the present findings has been made with respect to previous analyses and experimental observations.

Two-Dimensional Analysis of Conduction-Controlled Rewetting With Precursory Cooling

1976 ◽  
Vol 98 (3) ◽  
pp. 407-413 ◽  
Author(s):  
S. S. Dua ◽  
C. L. Tien
Keyword(s):  
Heat Transfer ◽  
Heat Transfer Coefficient ◽  
Transfer Coefficient ◽  
Dimensional Analysis ◽  
Vertical Surface ◽  
Dimensionless Temperature ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
One Dimensional ◽  
The One

This paper presents a two-dimensional analysis of the effect of precursory cooling on conduction-controlled rewetting of a vertical surface, whose initial temperature is higher than the sputtering temperature. Precursory cooling refers to the cooling caused by the droplet-vapor mixture in the region immediately ahead of the wet front, and is described mathematically by two dimensionless constants which characterize its magnitude and the region of influence. The physical model developed to account for precursory cooling consists of an infinitely extended vertical surface with the dry region ahead of the wet front characterized by an exponentially decaying heat flux and the wet region behind the moving film-front associated with a constant heat transfer coefficient. Apart from the two dimensionless constants describing the extent of precursory cooling, the physical problem is characterized by three dimensionless groups: the Peclet number or the dimensionless wetting velocity, the Biot number and a dimensionless temperature. Limiting solutions for large and small Peclet numbers have been obtained utilizing the Wiener-Hopf technique coupled with appropriate kernel substitutions. A semiempirical matching relation is then devised for the entire range of Peclet numbers. Existing experimental data with variable flow rates at atmospheric pressure are very closely correlated by the present model. Finally a comparison is drawn between the one-dimensional limit of the present analysis and the corresponding one-dimensional solution obtained by treating the dry region ahead of the wet front characterized by an exponentially decaying heat transfer coefficient.

Subjektive politische Strukturen in der Deutschschweiz

2001 ◽  
Vol 60 (3) ◽  
pp. 192-201 ◽  
Author(s):  
Wolfgang Marx ◽  
Linda Stähli
Keyword(s):  
Political Parties ◽  
Multidimensional Scaling ◽  
Nonmetric Multidimensional Scaling ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
One Dimensional ◽  
Sorting Data ◽  
Association Data

This article attempts to explore, via nonmetric multidimensional scaling (NMDS), the subjective structure of Swiss political parties. Free associations and hierarchical sortings of students are analyzed. The association data is described best by a one-dimensional MINISSA-solution showing the dimension left-right. The sorting data leads to a two-dimensional solution showing the dimension left-right and radicality. Differences and similarities to previous studies in Germany ( Marx & Läge, 1995 ) are discussed.

Numerical and Analytical Solutions for Two-Dimensional Gas Distribution in Gas-Loaded Heat Pipes

1989 ◽  
Vol 111 (3) ◽  
pp. 598-604 ◽  
Author(s):  
P. F. Peterson ◽  
C. L. Tien
Keyword(s):  
Heat Pipes ◽  
Computational Effort ◽  
Two Dimensional ◽  
Dimensional Model ◽  
Gas Distribution ◽  
Dimensional Solution ◽  
Integral Analysis ◽  
One Dimensional ◽  
Wide Range ◽  
Design Calculations

This work presents a two-dimensional axisymmetric diffusion model for the non-condensable gas distribution in gas-loaded heat pipes and thermosyphons. The new model, based on an integral analysis, has major advantages over existing, computationally time consuming, two-dimensional models. It has equal accuracy while using only the computational effort required for the cruder one-dimensional model, and also includes the effects of wall conduction and spatial variation of the condenser heat transfer coefficient. To simplify design calculations further an analytic two-dimensional solution is established, which gives excellent results over a wide range of parameters.

Split Mode Method for the Elliptic 2D Sine-Gordon Equation: Application to Josephson Junction in Overlap Geometry

1998 ◽  
Vol 09 (02) ◽  
pp. 301-323 ◽  
Author(s):  
Jean-Guy Caputo ◽  
Nikos Flytzanis ◽  
Yuri Gaididei ◽  
Irene Moulitsa ◽  
Emmanuel Vavalis
Keyword(s):  
Magnetic Field ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Josephson Junction ◽  
Gordon Equation ◽  
Splitting Method ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Dimensional Solution ◽  
One Dimensional

We introduce a new type of splitting method for semilinear partial differential equations. The method is analyzed in detail for the case of the two-dimensional static sine-Gordon equation describing a large area Josephson junction with overlap current feed and external magnetic field. The solution is separated into an explicit term that satisfies the one-dimensional sine-Gordon equation in the y-direction with boundary conditions determined by the bias current and a residual which is expanded using modes in the y-direction, the coefficients of which satisfy ordinary differential equations in x with boundary conditions given by the magnetic field. We show by direct comparison with a two-dimensional solution that this method converges and that it is an efficient way of solving the problem. The convergence of the y expansion for the residual is compared for Fourier cosine modes and the normal modes associated to the static one-dimensional sine-Gordon equation and we find a faster convergence for the latter. Even for such large widths as w=10 two such modes are enough to give accurate results.

scholarly journals Finite amplitude thermal convection with spatially modulated boundary temperatures

1995 ◽  
Vol 449 (1937) ◽  
pp. 459-478 ◽  
Keyword(s):  
Stability Analysis ◽  
Thermal Convection ◽  
Fluid Layer ◽  
Perturbation Technique ◽  
Lower Boundary ◽  
Finite Amplitude ◽  
Resonant Wavelength ◽  
Two Dimensional ◽  
Dimensional Solution ◽  
One Dimensional

Finite amplitude thermal convection in a fluid layer between two horizontal walls with different fixed mean temperatures is considered when spatially modulated temperatures with amplitudes L 1 * and L u * are prescribed at the lower and upper walls, respectively. The nonlinear steady problem is solved by a perturbation technique, and the preferred mode of convection is determined by a stability analysis. In the case of a resonant wavelength excitation, regular or non-regular multi-modal pattern convection can be preferred for some ranges of L 1 * and L u *, provided the wave vectors for such patterns are contained in a certain subset of the wave vectors representing a linear combination of modulated upper and lower boundary temperatures. In the case of non-resonant wavelength excitation, a three (two) dimensional solution in the form of multi-modal (rolls) pattern convection can be preferred, even if the boundary modulations are one (two) or two (one) dimensional, provided the wavelengths of the modulations are not too small. Heat transported by convection can be enhanced by boundary modulations in some ranges of L 1 * and L u *.

The Flat Plate Fin of Constant Thickness, Straight Base, and Symmetrical Shape

2014 ◽  
Vol 136 (5) ◽  
Author(s):  
Alejandro Rivera-Alvarez ◽  
Juan C. Ordonez
Keyword(s):  
Flat Plate ◽  
Shape Factor ◽  
Temperature Fields ◽  
Constant Thickness ◽  
Two Dimensional ◽  
One Dimensional ◽  
Symmetrical Shape ◽  
Extended Surface ◽  
And Performance ◽  
Biot Numbers

A plate fin is an extended surface made from a plate. Classical longitudinal and radial fins are particular cases of plate fins with very simple shapes and no curvature. In this paper, the problem of a flat plate fin of constant thickness, straight base, and symmetrical shape given by a proposed power law is considered. Particular attention is paid to some basic shapes: rectangular, triangular, convex parabolic, concave parabolic, convergent trapezoidal, and divergent trapezoidal. One- and two-dimensional analyses are conducted for every shape and comparison of results is carried through the usage of a proposed shape factor. Beyond shape, temperature fields and performance for the considered plate fins are shown to be dependent on a set of three Biot numbers characterizing the ratio between conduction resistances through every direction and convection resistance at the fin surface. Effectiveness and shape factor are found to be hierarchically organized by an including-figure rule. For the rectangular, zero-tip, and convergent trapezoidal cases, effectiveness is limited by a maximum possible value of Bit-1/2, and two-dimensional effects are very small. For the divergent trapezoidal case instead, effectiveness can be larger than Bit-1/2, and one-dimensional over-estimation of the actual heat transfer can be substantially large.

Solids Conveying in a Single Screw Extruder; the Rôle of Gravity Forces

1973 ◽  
Vol 15 (2) ◽  
pp. 114-122 ◽  
Author(s):  
J. G. A. Lovegrove ◽  
J. G. Williams
Keyword(s):  
Theoretical Analysis ◽  
Two Dimensional ◽  
Screw Extruder ◽  
Dimensional Solution ◽  
One Dimensional ◽  
Single Screw ◽  
Gravity Forces ◽  
Single Screw Extruder ◽  
Dimensional Version

A theoretical analysis is given which illustrates the rôle of gravity forces in solids conveying. A one-dimensional solution for flow in an extruder channel is used to investigate the nature of the solution and a more precise, two-dimensional, version is then developed.

scholarly journals Currents induced by long waves propagating towards a beach over a wavy bed

2000 ◽  
Vol 413 ◽  
pp. 317-343 ◽  
Author(s):  
H. KYOTOH ◽  
S. FUJII ◽  
D. V. TO
Keyword(s):  
Mass Flux ◽  
Wave Motion ◽  
Shallow Water Equation ◽  
Swash Zone ◽  
Two Dimensional ◽  
Maclaurin Series ◽  
Dimensional Solution ◽  
One Dimensional ◽  
Bed Slope ◽  
The One

For the understanding of longshore currents along a natural beach, the effects of bottom unevenness are considered to be important, especially for the flow in the swash zone. Currents in the swash zone are strongly influenced by the bed slope because the effect of gravity overwhelms the effect of the depth change. In the present paper, we investigate these effects and focus on waves propagating from offshore over a flat ocean basin of constant depth to a beach with a sloping wavy bottom. The waves are incident at a small angle to the beach normal, and the bed slope in the alongshore direction is varied slowly. To simplify the problem, only cnoidal waves and solitary waves are considered and the bed level is varied sinusoidally in the longshore direction.A perturbation method is applied to the two-dimensional nonlinear shallow water equation (two-dimensional NLSWE) for the wave motion in order to generate a more simplified model of wave dynamics consisting of a one-dimensional NLSWE for the direction normal to the beach and an equation for the alongshore direction. The first equation, the one-dimensional NLSWE, is solved by Carrier & Greenspan's transformation. The solution of the second one is found by extending Brocchini & Peregrine's solution for a flat beach. Two methods for the solution of the one- dimensional NLSWE are introduced in order to get a solution applicable to large-amplitude swash motions, where the amplitude is comparable to the beach length. One is the Maclaurin expansion of the solution around the moving shoreline, and the other is Riemann's representation of the solution, which exactly satisfies the one-dimensional NLSWE and the boundary conditions. After doing a consistency check by confirming that Riemann's method, a numerical solution, agrees with the exact solution for an infinitely long, sloping beach, we assumed that the Maclaurin series solution can also describe wave motion in the swash zone properly not only for this model but also for our ‘wavy’, finite beach model.The solution obtained from the Maclaurin series is then plugged into the equation for the alongshore direction to calculate the shore currents induced by wave run-up and back-wash motions, where a ‘weakly two-dimensional solution’ is derived from geometrical considerations. The results show that since the water depth near the shoreline is comparable to the bed level fluctuations, the flow is strongly affected by the bed unevenness, leading to recognizable changes in shoreline movement and the time-averaged velocity and the mass flux of the flow in the swash zone. More specifically, the inhomogeneity of the alongshore mass flux generates offshore currents because of the continuity condition for the fluid mass.

scholarly journals Analysis of Soft Soil Consolidation with the Application of Prefabricated Vertical Drains with Preloading Method Using Fem / Analiza Konsolidacji Słabych Gruntów Przy Uzyciu Metody Elementów Skonczonych

2011 ◽  
Vol 57 (1) ◽  
pp. 17-25 ◽  
Author(s):  
K. Binder
Keyword(s):  
Finite Element Method ◽  
Soft Soil ◽  
Two Dimensional ◽  
Soil Consolidation ◽  
Dimensional Solution ◽  
Prefabricated Vertical Drains ◽  
Vertical Drains ◽  
One Dimensional ◽  
Preloading Method ◽  
Consolidation Time

Abstract The article presents the application of finite element method for estimating settlements of road embankments founded on the soil reinforced with vertical drains and preloading method. The idea of the method was the transition from the solution of one-dimensional consolidation proposed for two-dimensional solution, while maintaining the same consolidation time and comparison with results obtained from measurements settlements of road embankment which is a part of planned Gdansk Southern Ring Road near Przejazdowo site.

The Heat Balance Integral in Steady-State Conduction

1976 ◽  
Vol 98 (3) ◽  
pp. 466-470 ◽  
Author(s):  
A. A. Sfeir
Keyword(s):  
Heat Balance ◽  
Integral Method ◽  
Two Dimensional ◽  
One Dimensional ◽  
Heat Balance Integral Method ◽  
Extended Surfaces ◽  
Different Shapes ◽  
The One ◽  
Biot Numbers ◽  
The Heat Balance

The Heat Balance Integral Method is applied to solve for the heat flow and temperature distribution in extended surfaces of different shapes and boundary conditions. In most cases the analysis is found to be identical to the exact two-dimensional solutions at Biot numbers for which the one-dimensional analysis is almost 100 percent off. Other possible extensions of the method are briefly described.

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