A Closed-form Equation for Predicting the Hydraulic Conductivity of Unsaturated Soils

1980 ◽  
Vol 44 (5) ◽  
pp. 892-898 ◽  
Author(s):  
M. Th. van Genuchten
Author(s):  
S. Elhanafi ◽  
K. Farhang

This paper considers leakage in mechanical seals under hydrostatic operating condition. A contact model based on the Greenwood and Williamson contact of rough surfaces is developed for treating problems involving mechanical seals in which both the micron scale roughness of the seal face and its macro scale profile are used to obtain either a closed-form equation or a nonlinear equation relating mean plane separation to the mass flow rate. The equations involve the micron scale geometry of the rough surfaces and physical parameter of the seal and carriage. Under hydrostatic condition, it is shown that there is an approximate closed-form solution in which mass flow rate in terms of the mean plane separation, or alternatively, the mean plane separation in terms of the leakage mass flow rate is found. Equations pertaining to leakage in nominally flat seal macro profile is considered and closed form equation relating to leakage flow rate to pressure difference is obtained that contain macro and micron geometries of the seal.


2000 ◽  
Author(s):  
Arnoldo Garcia ◽  
Arnold Lumsdaine ◽  
Ying X. Yao

Abstract Many studies have been performed to analyze the natural frequency of beams undergoing both flexural and torsional loading. For example, Adam (1999) analyzed a beam with open cross-sections under forced vibration. Although the exact natural frequency equation is available in literature (Lumsdaine et al), to the authors’ knowledge, a beam with an intermediate mass and support has not been considered. The models are then compared with an approximate closed form solution for the natural frequency. The closed form equation is developed using energy methods. Results show that the closed form equation is within 2% percent when compared to the transcendental natural frequency equation.


2018 ◽  
Vol 21 (12) ◽  
pp. 1911-1922
Author(s):  
Yang Zhan ◽  
Gang Wu

This article proposes a new closed-form equation to determine the reduction factor for global buckling of concentrically loaded pultruded fiber-reinforced polymer struts based on the Ayrton–Perry formula and observed initial out-of-straightness of pultruded fiber-reinforced polymer members measured by other researchers, which makes the original solution recommended by Eurocode 3 easy to be used to predict the global buckling loads of doubly symmetric pultruded fiber-reinforced polymer members subjected to axial compression. The influence of the geometric imperfections of pultruded fiber-reinforced polymer profiles is considered in this new closed-form equation. Validation of the solution including the parameter of the reduction factor for global buckling of pultruded fiber-reinforced polymer columns is performed by comparison with published experimental evidence. In addition, compared with the five closed-form solutions available in the literature, this solution exhibits higher accuracy in predicting the global buckling capacity of concentrically loaded pultruded fiber-reinforced polymer struts with doubly symmetric cross sections. The solution implemented into the new reduction factor equation for global buckling of pultruded fiber-reinforced polymer members can be conveniently used by structural engineers at the preliminary engineering design stage for accurately assessing the reliability and safety of composite structures under concentric compressive loading.


1998 ◽  
Vol 35 (6) ◽  
pp. 1093-1100 ◽  
Author(s):  
J R McDougall ◽  
I C Pyrah

Transient responses to various infiltration events have been examined using an unsaturated flow model. Numerical simulations reveal a range of infiltration patterns which can be related to the ratio of infiltration rate to unsaturated hydraulic conductivity. A high value of this ratio reflects a prevailing hydraulic conductivity which cannot readily redistribute the newly infiltrated moisture. Moisture accumulates in the near-surface region before advancing down through the soil as a distinct wetting front. In contrast, low values of the ratio of rainfall to unsaturated hydraulic conductivity show minimal moisture accumulation, as the relatively small volumes of infiltrating moisture are readily redistributed through the soil profile.Key words: numerical modelling, infiltration, unsaturated soil, soil suction, groundwater.


Soil Research ◽  
1969 ◽  
Vol 7 (2) ◽  
pp. 79 ◽  
Author(s):  
AJ Peck

Air bubbles in soil water affect both hydraulic conductivity and moisture content at a given capillary potential. Consequently changes in the volume of entrapped air, which are not included in the specification of relationships between hydraulic conductivity, moisture content, and capillary potential, will affect all soil-water interactions. Current understanding of the process of air bubble entrapment during infiltration suggests that, in nature, significant air entrapment will often occur. It is shown that infiltrating water can dissolve only a very small volume of air, much less than the amount usually entrapped. Air bubbles in saturated soils are unstable since their pressure must exceed atmospheric, resulting in a diffusive flux of dissolved air from bubbles to menisci contacting the external atmosphere. However, stable bubbles are possible in unsaturated soils. Bubbles which are constrained by pore architecture to non-spherical shapes are usually stable, and spherical bubbles can be stable when the magnitude of the capillary potential exceeds about 3 bars. An approximate analysis of the characteristic time of bubble equilibration indicates that, in an example, it is of order 104 sec, but it may be greater or less by at least a factor 10. Since the equilibration time will be often at least as large as the period of significant soil temperature changes, it cannot be assumed that the entrapped air in a field soil is in an equilibrium state. In such circumstances unstable bubbles may be quasi-permanent. It is suggested that the slow growth of entrapped bubbles may account for the anomalously slow release of water observed in some outflow experiments. Changes of entrapped air volume may also account for the reported dependence of soil-water characteristics on the magnitude of the steps of capillary potential.


2007 ◽  
Vol 7 (1) ◽  
pp. 59-66 ◽  
Author(s):  
D.E. Rolston

The science of soil-water physics and contaminant transport in porous media began a little more than a century ago. The first equation to quantify the flow of water is attributed to Darcy. The next major development for unsaturated media was made by Buckingham in 1907. Buckingham quantified the energy state of soil water based on the thermodynamic potential energy. Buckingham then introduced the concept of unsaturated hydraulic conductivity, a function of water content. The water flux as the product of the unsaturated hydraulic conductivity and the total potential gradient has become the accepted Buckingham-Darcy law. Two decades later, Richards applied the continuity equation to Buckingham's equation and obtained a general partial differential equation describing water flow in unsaturated soils. For combined water and solute transport, it had been recognized since the latter half of the 19th century that salts and water do not move uniformly. It wasn't until the middle of the 20th century that scientists began to understand the complex processes of diffusion, dispersion, and convection and to develop mathematical formulations for solute transport. Knowledge on water flow and solute transport processes has expanded greatly since the early part of the 20th century to the present.


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