Cantera/Diffusion - Revision history

Admin: moved Cantera Outline/Diffusion to Cantera/Diffusion

2014-01-13T18:01:03Z

moved Cantera Outline/Diffusion to Cantera/Diffusion

← Older revision	Revision as of 18:01, 13 January 2014
(No difference)

Admin at 19:00, 12 January 2014

2014-01-12T19:00:46Z

← Older revision		Revision as of 19:00, 12 January 2014
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	\dfrac{ 1 - Y_k }{ \sum_{j \neq k}{K} \dfrac{ X_j }{ \mathbb{D}_{jk} } }		\dfrac{ 1 - Y_k }{ \sum_{j \neq k}{K} \dfrac{ X_j }{ \mathbb{D}_{jk} } }
	</math>		</math>

			[[Category:Cantera]]

Admin: /* Mixture-Averaged */

2014-01-12T19:00:26Z

Mixture-Averaged

← Older revision		Revision as of 19:00, 12 January 2014
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	<math>		<math>
	u_{~~k,mixavg mass diff~~} = - \dfrac{1}{X_k} D_{km} \dfrac{d X_k}{dx}		V_{km} = - \dfrac{1}{X_k} D_{km} \dfrac{d X_k}{dx}
	</math>		</math>

	<math>		<math>
	u_{~~k,mixavg mass diff~~} = - \dfrac{1}{X_k} D_{km} \nabla X_k		V_{km} = - \dfrac{1}{X_k} D_{km} \nabla X_k
	</math>		</math>

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	<math>		<math>
	u_{~~k,mixavg therm diff~~} = - \dfrac{D_k^T}{\rho Y_k} \dfrac{1}{T} \dfrac{dT}{dx}		V_{kT} = - \dfrac{D_k^T}{\rho Y_k} \dfrac{1}{T} \dfrac{dT}{dx}
	</math>		</math>

	<math>		<math>
	u_{~~k,mixavg therm diff~~} = - \dfrac{D_k^T}{\rho Y_k} \dfrac{1}{T} \nabla T		V_{kT} = - \dfrac{D_k^T}{\rho Y_k} \dfrac{1}{T} \nabla T
	</math>		</math>

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	<math>		<math>
	D_{km} = \dfrac{ 1 - Y_k }{ \sum_{j \neq k}{K} \dfrac{ X_j }{ \mathbb{D}_{jk}		D_{km} =
			\dfrac{ 1 - Y_k }{ \sum_{j \neq k}{K} \dfrac{ X_j }{ \mathbb{D}_{jk} } }
	</math>		</math>

Admin: /* Multicomponent */

2014-01-12T18:58:48Z

Multicomponent

← Older revision		Revision as of 18:58, 12 January 2014
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	<math>		<math>
	\mathbf{J}_k = J_k = \rho u_{k~~,diff~~}		\mathbf{J}_k = J_k = \rho V_{k}
	</math>		</math>

	There are two contributions to the diffusion velocity <math>u_{k~~,diff~~}</math>: the multicomponent diffusion velocity, and a contribution to diffusion velocity via thermal gradients:		where <math>V_{k}</math> is the diffusive flux velocity for species k.

			There are two contributions to the diffusion velocity <math>V_{k}</math>: the multicomponent diffusion velocity, and a contribution to diffusion velocity via thermal gradients:

	<math>		<math>
	~~u_{k,diff}~~ = u_{~~k,mass diff~~} + u_{~~k,therm diff~~}		V_k = V_{km} + V_{kT}
	</math>		</math>

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	<math>		<math>
	u_{~~k,multicomponent mass diff~~} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \dfrac{ d X_j }{dx}		V_{km} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \dfrac{ d X_j }{dx}
	</math>		</math>

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	<math>		<math>
	u_{~~k,multicomponent multi diff~~} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \nabla X_j		V_{km} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \nabla X_j
	</math>		</math>

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	<math>		<math>
	u_{~~k,multicomponent therm diff~~} - \dfrac{D_k^T }{ \rho Y_k } \dfrac{1}{T} \dfrac{dT}{dx}		V_{kT} - \dfrac{D_k^T }{ \rho Y_k } \dfrac{1}{T} \dfrac{dT}{dx}
	</math>		</math>

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	<math>		<math>
	u_{~~k,multicomponent therm diff~~} = - \dfrac{D_k^T }{ \rho Y_k } \dfrac{1}{T} \nabla T		V_{kT} = - \dfrac{D_k^T }{ \rho Y_k } \dfrac{1}{T} \nabla T
	</math>		</math>

Admin at 18:57, 12 January 2014

2014-01-12T18:57:13Z

← Older revision		Revision as of 18:57, 12 January 2014
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	Cantera treats diffusion using either a multicomponent, species-by-species diffusion rate, or using a mixture-averaged diffusion rate for each species.		Cantera treats diffusion using either a multicomponent, species-by-species diffusion rate, or using a mixture-averaged diffusion rate for each species.

	~~Taken from~~ information available at http://public.ca.sandia.gov/chemkin/docs/oppdif.pdf		More information and context is available at http://public.ca.sandia.gov/chemkin/docs/oppdif.pdf

	=Diffusitivites=		=Diffusitivites=

Admin at 18:56, 12 January 2014

2014-01-12T18:56:47Z

← Older revision		Revision as of 18:56, 12 January 2014
Line 1:		Line 1:
	Cantera treats diffusion using either a multicomponent, species-by-species diffusion rate, or using a mixture-averaged diffusion rate for each species.		Cantera treats diffusion using either a multicomponent, species-by-species diffusion rate, or using a mixture-averaged diffusion rate for each species.

			Taken from information available at http://public.ca.sandia.gov/chemkin/docs/oppdif.pdf

	=Diffusitivites=		=Diffusitivites=

	~~The diffusion velocity (which determines the magnitude of the~~ diffusive ~~flux vector) for each species~~		Notation for some diffusivities should be defined before describing models for multicomponent or mixture-averaged diffusive fluxes. Three diffusivities are defined:

	* <math>D_{kj}</math> - multicomponent species diffusivity		* <math>D_{kj}</math> - multicomponent species diffusivity
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	* <Math>\mathbb{D}_{jk}</math> - binary diffusivity		* <Math>\mathbb{D}_{jk}</math> - binary diffusivity

	The diffusive flux <math> \mathbf{J} </math> is proportional to species gradients.		The diffusive flux <math> \mathbf{J} </math> is modeled as being proportional to species gradients. The diffusivities are the proportionality constants of that relationship.

	=Multicomponent=		=Multicomponent=

Admin: Created page with "Cantera treats diffusion using either a multicomponent, species-by-species diffusion rate, or using a mixture-averaged diffusion rate for each species. =Diffusitivites= The dif..."

2014-01-12T18:53:45Z

Created page with "Cantera treats diffusion using either a multicomponent, species-by-species diffusion rate, or using a mixture-averaged diffusion rate for each species. =Diffusitivites= The dif..."

New page

Cantera treats diffusion using either a multicomponent, species-by-species diffusion rate, or using a mixture-averaged diffusion rate for each species.

=Diffusitivites=

The diffusion velocity (which determines the magnitude of the diffusive flux vector) for each species

* <math>D_{kj}</math> - multicomponent species diffusivity
* <math>D_{km}</math> - mixture-averaged diffusivity
* <Math>\mathbb{D}_{jk}</math> - binary diffusivity

The diffusive flux <math> \mathbf{J} </math> is proportional to species gradients.

=Multicomponent=

The multicomponent diffusive flux model assumes the diffusive flux <math> \mathbf{J} </math> is proportional to every species gradient. This means that each species has a distinct contribution to the diffusive flux from the gradient of each species in the mixture:

(Assuming domain/diffusion is one-dimensional)

<math>
\mathbf{J}_k = J_k = \rho u_{k,diff}
</math>

There are two contributions to the diffusion velocity <math>u_{k,diff}</math>: the multicomponent diffusion velocity, and a contribution to diffusion velocity via thermal gradients:

<math>
u_{k,diff} = u_{k,mass diff} + u_{k,therm diff}
</math>

These are defined (assuming the one dimension is denoted <math>x</math>) as:

<math>
u_{k,multicomponent mass diff} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \dfrac{ d X_j }{dx}
</math>

or more generally,

<math>
u_{k,multicomponent multi diff} = \dfrac{1}{X_k \overline{W}} \sum_{j=1}^{K} W_j D_{kj} \nabla X_j
</math>

and the thermal diffusion term is:

<math>
u_{k,multicomponent therm diff} - \dfrac{D_k^T }{ \rho Y_k } \dfrac{1}{T} \dfrac{dT}{dx}
</math>

or,

<math>
u_{k,multicomponent therm diff} = - \dfrac{D_k^T }{ \rho Y_k } \dfrac{1}{T} \nabla T
</math>

=Mixture-Averaged=

The mixture-averaged diffusive flux model assumes the diffusive flux for species k, <math>J_k</math>, depends only on the gradient of species k:

<math>
u_{k,mixavg mass diff} = - \dfrac{1}{X_k} D_{km} \dfrac{d X_k}{dx}
</math>

<math>
u_{k,mixavg mass diff} = - \dfrac{1}{X_k} D_{km} \nabla X_k
</math>

and the thermal diffusion term is:

<math>
u_{k,mixavg therm diff} = - \dfrac{D_k^T}{\rho Y_k} \dfrac{1}{T} \dfrac{dT}{dx}
</math>

<math>
u_{k,mixavg therm diff} = - \dfrac{D_k^T}{\rho Y_k} \dfrac{1}{T} \nabla T
</math>

The mixture-averaged diffusivities are computed from the binary diffusivities as follows:

<math>
D_{km} = \dfrac{ 1 - Y_k }{ \sum_{j \neq k}{K} \dfrac{ X_j }{ \mathbb{D}_{jk}
</math>