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Related Concept Videos

Diffusion01:12

Diffusion

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Diffusion is the passive movement of substances down their concentration gradients—requiring no expenditure of cellular energy. Substances, such as molecules or ions, diffuse from an area of high concentration to an area of low concentration in the cytosol or across membranes. Eventually, the concentration will even out, with the substance moving randomly but causing no net change in concentration. Such a state is called dynamic equilibrium, which is essential for maintaining overall...
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Diffusion is a type of passive transport. In passive transport, a substance tends to move from an area of high concentration to an area of low concentration until the concentration is equal across the space. For example, take the diffusion of substances through the air. When someone opens a perfume bottle in a room filled with people, the perfume is at its highest concentration in the bottle and is at its lowest at the edges of the room. The perfume vapor will diffuse, or spread away, from the...
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Dimensional Analysis01:27

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Dimensional analysis is a valuable technique in fluid mechanics for simplifying complex problems by reducing them into dimensionless groups. These groups capture the essential relationships between the variables involved, allowing researchers and engineers to analyze fluid flow without dealing with each variable individually. This approach reduces the number of independent variables, allowing for easier analysis and better understanding of physical phenomena.
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Behavior of Gas Molecules: Molecular Diffusion, Mean Free Path, and Effusion03:48

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Although gaseous molecules travel at tremendous speeds (hundreds of meters per second), they collide with other gaseous molecules and travel in many different directions before reaching the desired target. At room temperature, a gaseous molecule will experience billions of collisions per second. The mean free path is the average distance a molecule travels between collisions. The mean free path increases with decreasing pressure; in general, the mean free path for a gaseous molecule will be...
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Theories of Dissolution: The Danckwerts' Model and Interfacial Barrier Model01:09

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Various dissolution theories provide insight into the factors that influence the dissolution rate. Danckwerts' Model suggests that turbulence, rather than a stagnant layer, characterizes the dissolution medium at the solid-liquid interface. In this model, the agitated solvent contains macroscopic packets that move to the interface via eddy currents, facilitating the absorption and delivery of the drug to the bulk solution. The regular replenishment of solvent packets maintains the...
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Electric Field of a Non Uniformly Charged Sphere01:22

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Gauss's law states that the electric flux through any closed surface equals the net charge enclosed within the surface. This law is beneficial for determining the expressions for the electric field for a particular charge distribution if the electric flux is known.
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The Diffusion of Passive Tracers in Laminar Shear Flow
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Analysis of Finite Difference Discretization Schemes for Diffusion in Spheres with Variable Diffusivity.

Ashlee N Ford Versypt1, Richard D Braatz1

  • 1Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA.

Computers & Chemical Engineering
|February 3, 2015
PubMed
Summary
This summary is machine-generated.

Two numerical schemes for the diffusion equation in spherical coordinates were compared. One scheme is recommended for its stability and accuracy, particularly when diffusivity depends on position.

Keywords:
diffusionfinite difference methodmethod of linesspherical geometryvariable coefficient

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Area of Science:

  • Computational physics
  • Numerical analysis
  • Chemical engineering

Background:

  • The diffusion equation is crucial for modeling transport phenomena.
  • Accurate numerical approximations are essential for complex diffusion scenarios.
  • Variable diffusivity presents significant challenges in numerical simulations.

Purpose of the Study:

  • To present and analyze two finite difference discretization schemes for the diffusion equation in spherical coordinates.
  • To compare the performance of these schemes for various forms of variable diffusivity.
  • To identify a stable and accurate scheme for spatially-dependent diffusivity.

Main Methods:

  • Developed two finite difference discretization schemes for spatial derivatives.
  • Applied schemes to the diffusion equation in spherical coordinates with variable diffusivity.
  • Compared numerical solutions against analytical/semi-analytical solutions for five diffusivity cases.
  • Evaluated scheme stability and physical reasonableness.

Main Results:

  • Both schemes showed good agreement for constant, temporally-dependent, spatially-dependent, and concentration-dependent diffusivity.
  • For implicitly-defined, temporally- and spatially-dependent diffusivity, one scheme diverged, while the other remained stable and physically reasonable.
  • The stable scheme demonstrated superior performance in complex scenarios.

Conclusions:

  • Finite difference schemes exhibit varying performance with complex diffusivity functions.
  • A specific finite difference scheme is recommended for its stability and accuracy.
  • The recommended scheme is particularly advantageous for diffusion problems where diffusivity is position-dependent.