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Statements

Subject Item
dbr:Finite_volume_method
dbo:wikiPageWikiLink
dbr:Finite_volume_method_for_two_dimensional_diffusion_problem
Subject Item
dbr:Finite_volume_method_for_one-dimensional_steady_state_diffusion
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dbr:Finite_volume_method_for_two_dimensional_diffusion_problem
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dbr:Finite_volume_method_for_two_dimensional_diffusion_problem
rdfs:label
Finite volume method for two dimensional diffusion problem
rdfs:comment
The methods used for solving two dimensional Diffusion problems are similar to those used for one dimensional problems. The general equation for steady diffusion can be easily derived from the general transport equation for property Φ by deleting transient and convective terms where, is the Diffusion coefficient and is the Source term. A portion of the two dimensional grid used for Discretization is shown below: Using the divergence theorem, the equation can be rewritten as : Flux across the east face = Flux across the south face = Flux across the north face = = Where, and .
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dbc:Computational_fluid_dynamics
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44532469
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1123217524
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dbr:Diffusion dbr:Computational_fluid_dynamics dbr:Fick's_laws_of_diffusion n9:Cfd_graph.jpg dbr:Diffusion_equation dbr:Boundary_value_problem dbr:Control_volume dbr:Maxwell–Stefan_equation dbr:Heat_equation dbr:Polygon_mesh dbr:Discretization dbr:Convection–diffusion_equation dbr:Fokker–Planck_equation dbr:Finite_difference dbr:Flux dbc:Computational_fluid_dynamics dbr:Taylor_series
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n14:Cfd_graph.jpg?width=300
dbp:date
2012-07-13
dbp:url
n20:dirCFD.htm
dbo:abstract
The methods used for solving two dimensional Diffusion problems are similar to those used for one dimensional problems. The general equation for steady diffusion can be easily derived from the general transport equation for property Φ by deleting transient and convective terms where, is the Diffusion coefficient and is the Source term. A portion of the two dimensional grid used for Discretization is shown below: In addition to the east (E) and west (W) neighbors, a general grid node P , now also has north (N) and south (S) neighbors. The same notation is usedhere for all faces and cell dimensions as in one dimensional analysis. When the above equation is formally integrated over the Control volume, we obtain Using the divergence theorem, the equation can be rewritten as : This equation represents the balance of generation of the property φ in a Control volume and the fluxes through its cell faces. The derivatives can by represented as follows by using Taylor series approximation: Flux across the east face = Flux across the south face = Flux across the north face = Substituting these expressions in equation (2) we obtain When the source term is represented in linearized form ,this equation can be rearranged as, = This equation can now be expressed in a general discretized equation form for internal nodes, i.e., Where, The face areas in y two dimensional case are : and . We obtain the distribution of the property i.e. a given two dimensional situation by writing discretized equations of the form of equation (3) at each grid node of the subdivided domain. At the boundaries where the temperature or fluxes are known the discretized equation are modified to incorporate the boundary conditions. The boundary side coefficient is set to zero (cutting the link with the boundary) and the flux crossing this boundary is introduced as a source which is appended to any existing and terms. Subsequently the resulting set of equations is solved to obtain the two dimensional distribution of the property
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