Contaminant transport in one-dimensional single fractured media: semi-analytical solution for three-member decay chain with pulse and Heaviside input sources

2007 ◽  
Vol 21 (16) ◽  
pp. 2135-2143 ◽  
Author(s):  
David Ching-Fang Shih
Keyword(s):  
Analytical Solution ◽  
Contaminant Transport ◽  
Decay Chain ◽  
Fractured Media ◽  
One Dimensional

scholarly journals Analytical solution of one-dimensional Pennes’ bioheat equation

Open Physics ◽  
2020 ◽  
Vol 18 (1) ◽  
pp. 1084-1092
Author(s):  
Hongyun Wang ◽  
Wesley A. Burgei ◽  
Hong Zhou
Keyword(s):  
Analytical Solution ◽  
Radiofrequency Energy ◽  
Bioheat Equation ◽  
Surface Cooling ◽  
One Dimensional ◽  
The Fourier Transform ◽  
The One ◽  
Parameter Values ◽  
Mathematical Derivation ◽  
Effect Of Surface

Abstract Pennes’ bioheat equation is the most widely used thermal model for studying heat transfer in biological systems exposed to radiofrequency energy. In their article, “Effect of Surface Cooling and Blood Flow on the Microwave Heating of Tissue,” Foster et al. published an analytical solution to the one-dimensional (1-D) problem, obtained using the Fourier transform. However, their article did not offer any details of the derivation. In this work, we revisit the 1-D problem and provide a comprehensive mathematical derivation of an analytical solution. Our result corrects an error in Foster’s solution which might be a typo in their article. Unlike Foster et al., we integrate the partial differential equation directly. The expression of solution has several apparent singularities for certain parameter values where the physical problem is not expected to be singular. We show that all these singularities are removable, and we derive alternative non-singular formulas. Finally, we extend our analysis to write out an analytical solution of the 1-D bioheat equation for the case of multiple electromagnetic heating pulses.

Building Vertical Walls by Deposition of Molten Metal Droplets

2005 ◽  
Author(s):  
M. Fang ◽  
S. Chandra ◽  
C. B. Park
Keyword(s):  
Surface Layer ◽  
Analytical Solution ◽  
Substrate Temperature ◽  
Layer By Layer ◽  
Droplet Temperature ◽  
Droplet Generator ◽  
One Dimensional ◽  
Metallurgical Bonding ◽  
Experimental Parameters ◽  
Vertical Walls

Experiments were conducted to determine conditions under which good metallurgical bonding was achieved in vertical walls composed of multiple layers of droplets that were fabricated by depositing tin droplets layer by layer. Molten tin droplets (0.75 mm diameter) were deposited using a pneumatic droplet generator on an aluminum substrate. The primary parameters varied in experiments were those found to most affect bonding between droplets on different layers: droplet temperature (varied from 250°C to 325°C) and substrate temperature (varied from 100°C to 190°C). Considering the cooling rate of droplet is much faster than the deposition rate previous deposition layer cooled down too much that impinging droplets could only remelt a thin surface layer after impact. Assuming that remelting between impacting droplets and the previous deposition layer is a one-dimensional Stefan problem with phase change an analytical solution can be found and applied to predict the minimum droplet temperature and substrate temperature required for local remelting. It was experimentally confirmed that good bonding at the interface of two adjacent layers could be achieved when the experimental parameters were such that the model predicted remelting.

Analysis of a model for contaminant transport in fractured media in the presence of colloids

1995 ◽  
Vol 165 (1-4) ◽  
pp. 261-281 ◽  
Author(s):  
Assem Abdel-Salam ◽  
Constantinos V. Chrysikopoulos
Keyword(s):  
Contaminant Transport ◽  
Fractured Media

Channeling Flow and Solute Transport in a Single Fracture

2013 ◽  
Vol 316-317 ◽  
pp. 632-635
Author(s):  
Ye Fei Tan ◽  
Zhi Fang Zhou ◽  
Shi Qiang Wu ◽  
Xing Hua Xie ◽  
Bo Ning
Keyword(s):  
Finite Element Method ◽  
Drinking Water ◽  
Breakthrough Curve ◽  
Laboratory Experiments ◽  
Fractured Media ◽  
Single Fracture ◽  
The Finite Element Method ◽  
Dispersive Equation ◽  
One Dimensional ◽  
Channeling Flow

Groundwater in fractured media plays an important role in drinking water supply, and the understanding of its principle mechanisms is essential for securing the groundwater exploring and utilization. In this paper, a novel conceptual fracture model was presented on the basis of the reality of channeling flow in natural fractures and laboratory experiments were conducted for the purpose of getting a better understanding of the step-like breakthrough curve (BTC). Experimental results were fitted with convective dispersive equation (CDE) and compared with those of the finite element method (FEM) models. Results showed that the traditional one-dimensional CDE was invalid in the fitting of a step-like BTC and needed to be improved.

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