Analytical solution to an ionomer-filled agglomerate problem in 1D. More...

#include <agglomerate_ionomer_sun.h>

Inheritance diagram for FuelCellShop::MicroScale::IonomerAgglomerateSun:

Collaboration diagram for FuelCellShop::MicroScale::IonomerAgglomerateSun:

Public Member Functions
virtual SolutionMap	compute_current ()
	Main function of the class used to compute the current over the whole agglomerate at the local operating conditions. More...

virtual std::vector< double >	compute_derivative_current ()
	Function to compute the derivative of the current density at the local operating conditions;. More...

virtual std::string	get_name ()
	Return name of class instance, i.e. More...

virtual double	aux_volume_fraction ()
	Returns extra contribution to volume of layer. More...

Public Member Functions inherited from FuelCellShop::MicroScale::AgglomerateBase
virtual	~AgglomerateBase ()

virtual void	print_properties ()
	Print out key agglomerate information (name, radius, film thickness, porosity). More...

virtual bool	has_derivatives ()
	Returns true if the class instance can calculate current density derivatives. More...

virtual void	make_thread_safe (ParameterHandler &param, unsigned int thread_index)

virtual void	set_solution (const std::map< VariableNames, SolutionVariable > &, const VariableNames &, const int &)
	Function for setting the solution map(reactant concentration, phi_s, phi_m, etc.). More...

Public Member Functions inherited from FuelCellShop::MicroScale::MicroScaleBase
virtual	~MicroScaleBase ()
	Destructor. More...

Static Public Attributes
static const std::string	concrete_name

Protected Member Functions
virtual double	get_film_thickness ()

virtual double	get_radius ()

virtual void	set_structure ()

virtual boost::shared_ptr < FuelCellShop::MicroScale::MicroScaleBase >	create_replica ()
	This member function is used to create an object of MicroScaleBase. More...

	IonomerAgglomerateSun ()
	Constructors. More...

	IonomerAgglomerateSun (std::string concrete_name)

virtual void	declare_parameters (ParameterHandler &param) const

virtual void	initialize (ParameterHandler &param)

Protected Member Functions inherited from FuelCellShop::MicroScale::AgglomerateBase
void	_initialize_film_porosity ()

	AgglomerateBase ()

Protected Member Functions inherited from FuelCellShop::MicroScale::MicroScaleBase
void	set_layer (FuelCellShop::Layer::MultiScaleCL< deal_II_dimension > *_layer)

	MicroScaleBase ()

Protected Member Functions inherited from FuelCellShop::MicroScale::SphericalAgglomerateGeometry
	SphericalAgglomerateGeometry ()

virtual double	compute_thickness_agg (FuelCellShop::Layer::MultiScaleCL< deal_II_dimension > *layer)
	Member function to compute the thickness of the agglomerate thin film based on the radius and structure. More...

virtual double	compute_epsilon_agg (FuelCellShop::Layer::MultiScaleCL< deal_II_dimension > *layer)
	Compute the volume fraction of nafion inside the agglomerate given the catalyst layer ionomer volume fraction , $\epsilon_N$ ; size of the agglomerate, $r_{agg}$ ; and, porosity inside the agglomerate, $\epsilon_{agg}$ , the necessary ionomer thin film thickness is computed such that the correct volume fractions are obtained. More...

virtual double	compute_epsilon_N (const double delta_agg, const double thickness_agg) const
	Function used to compute the amount of electrolyte in the catalyst layer. More...

virtual double	compute_depsilonN_dthickness (const double thickness_agg) const
	Function to compute $\frac{\partial \epsilon_N^{cat}}{\partial thickness_{agg}}$ This function should only be used to calculate the thin film thickness using Newton's method. More...

virtual double	compute_depsilonN_depsilon_agg (FuelCellShop::Layer::MultiScaleCL< deal_II_dimension > *layer) const
	Function to compute $\frac{\partial \epsilon_N^{cat}}{\partial \epsilon_{agg}}$ This function should only be used to calculate the thin film thickness using Newton's method. More...

virtual double	compute_n (FuelCellShop::Layer::MultiScaleCL< deal_II_dimension > *layer) const
	Member function to compute the number of agglomerates. More...

virtual double	compute_dn_depsilon_agg (FuelCellShop::Layer::MultiScaleCL< deal_II_dimension > *layer) const
	Compute the derivative of n with respect to the porosity of the agglomerate. More...

Static Protected Attributes
static IonomerAgglomerateSun const *	PROTOTYPE

Private Member Functions
void	check_kinetics ()

double	compute_Er (const double k_c, const double D)
	Function to compute the effectiveness of the agglomerate core. More...

double	compute_dEr (const double k_c, const double dk_c, const double D)
	Function to compute the derivative of the effectiveness of the agglomerate core. More...

Private Attributes
bool	checked_kinetics

double	epsilon_V
	Catalyst layer porosity. More...

Additional Inherited Members
Static Public Member Functions inherited from FuelCellShop::MicroScale::MicroScaleBase
static void	declare_MicroScale_parameters (ParameterHandler &param)
	Function used to declare all the data necessary in the parameter files for all MicroScale children. More...

static boost::shared_ptr < FuelCellShop::MicroScale::MicroScaleBase >	create_MicroStructure (ParameterHandler &param, FuelCellShop::Layer::MultiScaleCL< deal_II_dimension > *layer)
	Function used to select the appropriate MicroScale type as specified in the ParameterHandler under line. More...

Protected Types inherited from FuelCellShop::MicroScale::AgglomerateBase
typedef FuelCellShop::Layer::MultiScaleCL < deal_II_dimension > ::Properties	CLPropNames

typedef std::map< CLPropNames, double >	CL_Properties

Protected Types inherited from FuelCellShop::MicroScale::MicroScaleBase
typedef std::map< std::string, MicroScaleBase * >	_mapFactory
	This object is used to store all objects of type MicroScaleBase. More...

Protected Types inherited from FuelCellShop::MicroScale::SphericalAgglomerateGeometry
typedef FuelCellShop::Layer::MultiScaleCL < deal_II_dimension > ::Properties	CLPropNames

typedef std::map< CLPropNames, double >	CL_Properties

Static Protected Member Functions inherited from FuelCellShop::MicroScale::MicroScaleBase
static _mapFactory *	get_mapFactory ()
	This member function is used to create an object of type MicroScaleBase. More...

Protected Attributes inherited from FuelCellShop::MicroScale::AgglomerateBase
boost::shared_ptr < FuelCellShop::Material::CatalystBase >	catalyst
	Boost shared pointer to catalyst object. More...

boost::shared_ptr < FuelCellShop::Material::PolymerElectrolyteBase >	electrolyte
	Boost shared pointer to electrolyte object. More...

boost::shared_ptr < FuelCellShop::Kinetics::BaseKinetics >	kinetics
	Boost shared pointer to kinetics object. More...

std::map< VariableNames, SolutionVariable >	solutions
	Member data for storing solutions. More...

std::vector< VariableNames >	sol_names

VariableNames	reactant

VariableNames	tempReactantName

int	sol_index

double	permittivity_0
	Constants. More...

double	F

double	R

double	P

double	AV

double	c_R

bool	has_derivatives_

double	c_H

double	phi_M

double	phi_S

double	interface

double	D_R_N

double	H_R_N
	Henry's Constant for primary reactant in nafion, [Pa cm^3/mol]. More...

std::string	fixed_agg_variable

Protected Attributes inherited from FuelCellShop::MicroScale::MicroScaleBase
FuelCellShop::Layer::MultiScaleCL < deal_II_dimension > *	layer

Protected Attributes inherited from FuelCellShop::MicroScale::SphericalAgglomerateGeometry
double	pi

double	r_agg

double	delta_agg

double	epsilon_agg

double	n_agg

Detailed Description

Analytical solution to an ionomer-filled agglomerate problem in 1D.

This class implements the analytical ionomer-filled agglomerate first proposed by:

W. Sun, B. A. Peppley, K. Karan, An improved two-dimensional agglomerate cathode model to study the influence of catalyst layer structural parameters, Electrochimica Acta 50 (16-17):3347-3358, (2005).

and later implemented in openFCST in order to provide the results obtained in:

M. Secanell, K. Karan, A. Suleman and N. Djilali, "Multi-Variable Optimization of PEMFC Cathodes using an Agglomerate Model ", Electrochimica Acta, 52(22):6318-6337, 2007.

The data files for this article can be found in fcts/data/articles/Secanell_EA07_MultiVariable_Optimization_PEMFC_Cathodes_Agglomerate_Model/. You can find three folders there one for analyzing a single case, one for an IV curve and one for running an optimization case as discussed in the article.

Theory

The volumetric current density is

$\nabla \cdot \vec{i} = 4F \frac{p_{tot}x_{O_2}}{H_{O_2,N}}\left(\frac{1}{E_r k_c (1-\epsilon^{cl}_V)} + \frac{(r_{agg} + \delta_{agg})\delta_{agg}}{a_{agg}r_{agg}D_{O_2, N}} \right)^{-1}$

where

$k_c = \frac{A_v i^{ref}_0}{4F(1-\epsilon^{cl}_V)c^{ref}_{O_2}} \exp(-\frac{\alpha_c F}{RT}(\phi_s - \phi_m))$

and $ k_c $ is the reaction rate constant, $s^{-1}$ , $i^{ref}_0$ is the exchange current density, [ $A \cdot cm^{-2}$ ], $ R $ is the gas constant, 8.315 $J \cdot K^{-1} \cdot mol^{-1}$ ], $ T $ is the temperature, [K], $\gamma$ is the coefficient in Tafel equation, [-], $ F $ is the Faraday constant, 96493 [ $C \cdot mol$ ], $\alpha_c$ is the transfer coefficient, [-], $c^{ref}_{O_2}$ is the reference oxygen concentration, [ $mol \cdot cm^{-3}$ ], the term $1-\epsilon^{cat}_V$ is used to transform the active area in the catalyst layer, $ A_v $ to an active area inside the agglomerate and the effectiveness factor is given by

$E_r = \frac{1}{\phi_L}\left( \frac{1}{\tanh(3\phi_L)} - \frac{1}{3\phi_L} \right)$

where $ E_r $ is the effectiveness factor, [-] and the Thiele's modulus for a spherical agglomerate is given by

$\phi_L = \frac{r_{agg}}{3} \sqrt{\frac{k_c}{D^{eff}}}$

where $D^{eff}$ is the effective oxygen diffusion coefficient inside the agglomerate, [ $cm^2\cdot s^{-1}$ ], $r_{agg}$ is the radius of the spherical agglomerate and $D^{eff}$ is the effective oxygen diffusion coefficient inside the agglomerate.

In these equations, there are several parameters that need to be obtained: $H_{O_2,N}$ , $a_{agg}$ , $D_{O_2,N}$ , $ A_v $ , $i^{ref}_0$ , $c^{ref}_{O_2}$ and $D^{eff}$ . Parameters $H_{O_2,N}$ , $D_{O_2, N}$ , $i^{ref}_0$ and $c^{ref}_{O_2}$ are input parameters to the model and are obtained from transport and electrochemical data. In the code, they are specified by the type of kinetics and catalyst used. Regarding the type of kinetic model, ONLY TAFEL can be used in this case.

Parameters $a_{agg}$ , $ A_v $ and $D^{eff}$ depend on the composition of the catalyst layer.

The parameter $a_{agg}$ is defined as the ratio between the effective surface area usable to dissolve oxygen into the agglomerate and the catalyst layer volume. This value can be related to the catalyst layer structure by

$a_{agg} = n 4 \pi (r_{agg} + \delta_{agg})^2 \epsilon^{cl}_V$

where $\delta_{agg}$ is the thin electrolyte film surrounding the agglomerate, [ $\mu m$ ], $ n $ is the number of agglomerates per unit volume, the term $4 \pi (r_{agg} + \delta_{agg})^2$ is the surface of a single agglomerate and finally, $\epsilon^{cl}_V$ is the catalyst layer porosity.

For more information see:

M. Secanell, K. Karan, A. Suleman and N. Djilali, "Multi-Variable Optimization of PEMFC Cathodes using an Agglomerate Model ", Electrochimica Acta, 52(22):6318-6337, 2007.

Author: M. Secanell 2004-13

Constructor & Destructor Documentation

FuelCellShop::MicroScale::IonomerAgglomerateSun::IonomerAgglomerateSun ( )

protected

Constructors.

FuelCellShop::MicroScale::IonomerAgglomerateSun::IonomerAgglomerateSun ( std::string concrete_name )

protected

Member Function Documentation

virtual double FuelCellShop::MicroScale::IonomerAgglomerateSun::aux_volume_fraction ( )

inlinevirtual

Returns extra contribution to volume of layer.

Implements FuelCellShop::MicroScale::MicroScaleBase.

void FuelCellShop::MicroScale::IonomerAgglomerateSun::check_kinetics ( )

private

virtual SolutionMap FuelCellShop::MicroScale::IonomerAgglomerateSun::compute_current ( )

virtual

Main function of the class used to compute the current over the whole agglomerate at the local operating conditions.

Implements FuelCellShop::MicroScale::MicroScaleBase.

double FuelCellShop::MicroScale::IonomerAgglomerateSun::compute_dEr	(	const double	k_c,
		const double	dk_c,
		const double	D
	)

private

Function to compute the derivative of the effectiveness of the agglomerate core.

virtual std::vector<double> FuelCellShop::MicroScale::IonomerAgglomerateSun::compute_derivative_current ( )

virtual

Function to compute the derivative of the current density at the local operating conditions;.

Reimplemented from FuelCellShop::MicroScale::AgglomerateBase.

double FuelCellShop::MicroScale::IonomerAgglomerateSun::compute_Er	(	const double	k_c,
		const double	D
	)

private

Function to compute the effectiveness of the agglomerate core.

virtual boost::shared_ptr<FuelCellShop::MicroScale::MicroScaleBase> FuelCellShop::MicroScale::IonomerAgglomerateSun::create_replica ( )

inlineprotectedvirtual

This member function is used to create an object of MicroScaleBase.

Warning: This class MUST be redeclared in every child.

Implements FuelCellShop::MicroScale::MicroScaleBase.

virtual void FuelCellShop::MicroScale::IonomerAgglomerateSun::declare_parameters ( ParameterHandler & param ) const

inlineprotectedvirtual

Reimplemented from FuelCellShop::MicroScale::AgglomerateBase.

References concrete_name, and FuelCellShop::MicroScale::AgglomerateBase::declare_parameters().

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virtual double FuelCellShop::MicroScale::IonomerAgglomerateSun::get_film_thickness ( )

inlineprotectedvirtual

Implements FuelCellShop::MicroScale::AgglomerateBase.

References FuelCellShop::MicroScale::SphericalAgglomerateGeometry::delta_agg.

virtual std::string FuelCellShop::MicroScale::IonomerAgglomerateSun::get_name ( )

inlinevirtual

Return name of class instance, i.e.

concrete name.

Implements FuelCellShop::MicroScale::MicroScaleBase.

References concrete_name.

virtual double FuelCellShop::MicroScale::IonomerAgglomerateSun::get_radius ( )

inlineprotectedvirtual

Implements FuelCellShop::MicroScale::AgglomerateBase.

References FuelCellShop::MicroScale::SphericalAgglomerateGeometry::r_agg.

virtual void FuelCellShop::MicroScale::IonomerAgglomerateSun::initialize ( ParameterHandler & param )

inlineprotectedvirtual

Reimplemented from FuelCellShop::MicroScale::AgglomerateBase.

References concrete_name, and FuelCellShop::MicroScale::AgglomerateBase::initialize().

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virtual void FuelCellShop::MicroScale::IonomerAgglomerateSun::set_structure ( )

protectedvirtual

Implements FuelCellShop::MicroScale::MicroScaleBase.

Member Data Documentation

bool FuelCellShop::MicroScale::IonomerAgglomerateSun::checked_kinetics

private

const std::string FuelCellShop::MicroScale::IonomerAgglomerateSun::concrete_name

static

Referenced by declare_parameters(), get_name(), and initialize().

double FuelCellShop::MicroScale::IonomerAgglomerateSun::epsilon_V

private

Catalyst layer porosity.

IonomerAgglomerateSun const* FuelCellShop::MicroScale::IonomerAgglomerateSun::PROTOTYPE

staticprotected

The documentation for this class was generated from the following file:

agglomerate_ionomer_sun.h

Public Member Functions

Static Public Attributes

Protected Member Functions

Static Protected Attributes

Private Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Theory

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation