Mathematical Modeling of a Solid–Liquid Mixture with Mass Exchange Between Constituents

2006 ◽  
Vol 11 (6) ◽  
pp. 575-595 ◽  
Author(s):  
L. Fusi ◽  
A. Farina ◽  
D. Ambrosi
Keyword(s):  
Mathematical Modeling ◽  
Liquid Mixture ◽  
Solid Phase ◽  
Deformation Gradient ◽  
Elastic Solid ◽  
Solid Deformation ◽  
Solid Stress ◽  
Phase Deformation ◽  
Mass Conversion ◽  
Solid Liquid

The mechanical behavior of a mixture composed by an elastic solid and a fluid that exchange mass is investigated. Both the liquid flow and the solid deformation depend on how the solid phase has increased (diminished) its mass, i.e. on the mass conversion between constituents. The model is developed introducing a decomposition of the solid phase deformation gradient. In particular, exploiting the criterion of maximization of the rate of entropy production, we determine an explicit evolution equation for the so-called growth tensor which involves directly the solid stress tensor. An example of a possible choice of the constitutive functions is also presented.

Solubility of Stearic Acid in Ethanol, 1-Propanol, 2-Propanol, L-Butanol, Acetone, Methylene Chloride, Ethyl Acetate and 95% Ethanol from (293 to 315) K

2012 ◽  
Vol 550-553 ◽  
pp. 71-74
Author(s):  
Wen Ju Liu ◽  
Chen Sun ◽  
Pei Xia Zhao ◽  
Shao Feng Wang
Keyword(s):  
Stearic Acid ◽  
Ethyl Acetate ◽  
Methylene Chloride ◽  
Liquid Mixture ◽  
Solid Phase ◽  
Synthetic Method ◽  
Laser Monitoring ◽  
Solid Liquid ◽  
Semi Empirical ◽  
Laser Monitoring Observation Technique

Solubility of stearic acid in ethanol, 1-propanol, 2-propanol,l-butanol, acetone, methylene chloride, ethyl acetate and 95% ethanol was experimentally determined by the synthetic method from (293 to 315)K. The laser monitoring observation technique was used to determine the disappearance of the solid phase in a solid + liquid mixture. The solubility of stearic acid in methylene chloride was found to be the highest, and the value in 95% ethanol was the lowest. Results are correlated by semi-empirical equations, which show a good fit to the experimental data.

Mathematical Modeling of Betanin Extraction from Red Beet (Beta vulgaris L.) by Solid–Liquid Method

2015 ◽  
Vol 11 (1) ◽  
pp. 17-22 ◽  
Author(s):  
Honggao Xu ◽  
Qiang Peng ◽  
Fang Yuan ◽  
Yanxiang Gao
Keyword(s):  
Mathematical Modeling ◽  
Effective Diffusion ◽  
Spherical Model ◽  
Extraction Process ◽  
Extraction Temperature ◽  
Optimal Parameters ◽  
Practical Application ◽  
Sulfurous Acid ◽  
Red Beet ◽  
Solid Liquid

Abstract Effective diffusion coefficient (D-value) is an important parameter for the extraction process. In this study, a cylindrical model to estimate the D-value of betanin extraction from red beet roots was developed. The influence of extraction temperature, pH, cylindrical thickness and type of acids on the D-value was also investigated. The optimal parameters of betanin extraction were 30°C, pH 4.0 adjusted with sulfurous acid or acetic acid. D-values under the above conditions were 18.95×10−11 m2/s and 17.87×10−11 m2/s, respectively. The result showed that D-value elevated with the increase of cylindrical thickness, which was consistent with earlier conclusion from spherical model. The modeling may be useful for the investigation into extraction process and practical application.

On the Natural Lubrication of Synovial Joints: Normal and Degenerate

1977 ◽  
Vol 99 (2) ◽  
pp. 163-172 ◽  
Author(s):  
Joseph M. Mansour ◽  
Van C. Mow
Keyword(s):  
Articular Cartilage ◽  
Solid Phase ◽  
Articular Surface ◽  
Moving Load ◽  
Frictional Resistance ◽  
Elastic Solid ◽  
Surface Load ◽  
Two Phase ◽  
Tissue Properties ◽  
Synovial Joints

Fluid flow and mass transport mechanisms associated with articular cartilage function are important biomechanical processes of normal and pathological synovial joints. A three-layer permeable, two-phase medium of an incompressible fluid and a linear elastic solid are used to model the flow and deformational behavior of articular cartilage. The frictional resistance of the relative motion of the fluid phase with respect to the solid phase is given by a linear diffusive dissipation term. The subchondral bony substrate is represented by an elastic solid. The three-layer model of articular cartilage is chosen because of the known histological, ultrastructural, and biomechanical variations of the tissue properties. The calculated flow field shows that for material properties of normal healthy articular cartilage the tissue creates a naturally lubricated surface. The movement of the interstitial fluid at the surface is circulatory in manner, being exuded in front and near the leading half of the moving surface load and imbibed behind and near the trailing half of the moving load. The flow fields of healthy tissues are capable of sustaining a film of fluid at the articular surface whereas pathological tissues cannot.

Unusual liquid-solid transition effects in a methoxypentamethylflavan

1958 ◽  
Vol 243 (1235) ◽  
pp. 518-533 ◽  
Keyword(s):  
Solid Phase ◽  
Polarized Light ◽  
Melting Process ◽  
X Rays ◽  
Melting Range ◽  
Crystalline Materials ◽  
External Conditions ◽  
Transition Effects ◽  
Solid Liquid ◽  
Crystal Boundaries

The solid phase of the compound 2'-methoxy-2 : 4 : 4 : 7 : 4'- pentamethylflavan differs in many respects from normally crystalline materials, yet is not amorphous. It tends to form spherical masses which exhibit no regular crystal boundaries, even when seen under the microscope. The solid melts over a range of up to 3° C and the actual temperatures at which melting begins and ends vary according to the thermal treatment previously received by the specimen. The temperature at which the melt starts to solidify on cooling is always several degrees below the melting range; this is not due to normal supercooling. Solidification from the melt presents several other interesting features, including some rhythmic effects. Variations in the external conditions during solidification can give rise to three superficially different forms of the solid phase. The solid-liquid and liquid-solid transitions have been followed by measurements of density, rigidity and dielectric constant, all of which give further indications of the diffuse nature of the melting process and the existence of hysteresis between melting and solidifica­tion. These effects recall the behaviour of some crystalline high polymers and examination of solid methoxypentamethylflavan by polarized light, X-rays and electron microscopy has revealed further analogies with such materials. It is tentatively concluded that the solid is composed of submicroscopic crystalline regions which are organized into larger spherulitic aggregates, but no definite explanations of the failure of the compound to form macroscopic crystals or of the similarities between it and polymers seem possible at present.

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