Class that gives the analytical solution to an ionomer-filled agglomerate problem in 1D. More...

#include <agglomerate_ionomer_analytical.h>

Inheritance diagram for FuelCellShop::MicroScale::IonomerAgglomerateAnalytical:

Collaboration diagram for FuelCellShop::MicroScale::IonomerAgglomerateAnalytical:

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...

	IonomerAgglomerateAnalytical ()
	Constructors. More...

	IonomerAgglomerateAnalytical (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 IonomerAgglomerateAnalytical const *	PROTOTYPE

Private Member Functions
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...

void	check_kinetics ()

Private Attributes
bool	checked_kinetics

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

Class that gives the analytical solution to an ionomer-filled agglomerate problem in 1D.

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

M. Moore, P. Wardlaw, P. Dobson, R. Spiteri, J. B and M. Secanell, "", Journal of the Electrochemical Society

Theory

The volumetric current density is

$\nabla \cdot \vec{i} = \frac{1}{1-\epsilon_V} 4F \bar{V}_{agg} \frac{P_{O_2}}{H_{O_2,N}}\left[\frac{1}{E_r k_c} + \frac{\delta_{agg}r_{agg}^2}{3\left(r_{agg}+\delta_{agg}\right)D_{O_2,N}}\right]^{-1}$

The effectiveness factor is obtained from the analytical solution of the oxygen transport equation on the agglomerate domain,

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

where $\phi_L$ is Thieles modulus, which characterizes the reaction-transport process for a given geometry. For a sphere, the characteristic length is $\frac{r_{agg}}{3}$ , so Thiele's modulus becomes

$\phi_L = \frac{r_{agg}}{3} \sqrt{\frac{k_c}{D^{eff}_{O_2}}}$

The term $\left(1-\varepsilon_V\right)\bar{V}_{agg}$ is an active area scaling factor. Typically, the active area for an electrode is given as the area per volume of catalyst layer. Since in the agglomerate model the platinum is only found in the core of the agglomerate, the active area has to be adjusted accordingly. Dividing by $\left(1-\varepsilon_V\right)$ , where $\epsilon_V$ is the porosity of the electrode, gives the active area of Pt ( $cm^2_{Pt}$ ) per volume of agglomerate ( $cm^3_{agg}$ ). Then, dividing by $\bar{V}_{agg}$ gives the active area of Pt ( $cm^2_{Pt}$ ) per volume of agglomerate core ( $cm^3_{agg, core}$ ). The variable ${V}_{agg}$ is defined as

$\bar{V}_{agg} = \frac{V_{agg}}{V_{tot}} = \frac{\frac{4\pi r_{agg}^3}{3}}{\frac{4\pi \left(r_{agg}+\delta_{agg}\right)^3}{3}} = \dfrac{r_{agg}^3}{\left(r_{agg}+\delta_{agg}\right)^3}$

is a scaling factor determined as the ratio of the volume of the agglomerate core to the volume of the entire agglomerate.

Author: P. Dobson, P. Wardlaw and M. Secanell 2009-13

Constructor & Destructor Documentation

FuelCellShop::MicroScale::IonomerAgglomerateAnalytical::IonomerAgglomerateAnalytical ( )

protected

Constructors.

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

protected

Member Function Documentation

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

inlinevirtual

Returns extra contribution to volume of layer.

Implements FuelCellShop::MicroScale::MicroScaleBase.

void FuelCellShop::MicroScale::IonomerAgglomerateAnalytical::check_kinetics ( )

private

virtual SolutionMap FuelCellShop::MicroScale::IonomerAgglomerateAnalytical::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::IonomerAgglomerateAnalytical::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::IonomerAgglomerateAnalytical::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::IonomerAgglomerateAnalytical::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::IonomerAgglomerateAnalytical::create_replica ( )

inlineprotectedvirtual

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

Implements FuelCellShop::MicroScale::MicroScaleBase.

virtual void FuelCellShop::MicroScale::IonomerAgglomerateAnalytical::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::IonomerAgglomerateAnalytical::get_film_thickness ( )

inlineprotectedvirtual

Implements FuelCellShop::MicroScale::AgglomerateBase.

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

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

inlinevirtual

Return name of class instance, i.e.

concrete name.

Implements FuelCellShop::MicroScale::MicroScaleBase.

References concrete_name.

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

inlineprotectedvirtual

Implements FuelCellShop::MicroScale::AgglomerateBase.

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

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

protectedvirtual

Implements FuelCellShop::MicroScale::MicroScaleBase.

Member Data Documentation

bool FuelCellShop::MicroScale::IonomerAgglomerateAnalytical::checked_kinetics

private

const std::string FuelCellShop::MicroScale::IonomerAgglomerateAnalytical::concrete_name

static

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

IonomerAgglomerateAnalytical const* FuelCellShop::MicroScale::IonomerAgglomerateAnalytical::PROTOTYPE

staticprotected

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

agglomerate_ionomer_analytical.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