pb_flow_battery

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Soluble Lead-Acid Redox Flow Battery

Introduction

In a redox flow battery electrochemical energy is stored as redox couples in the

electrolyte, which is stored in tanks outside the electrochemical cell. During operation, electrolyte is pumped through the cell and, due to the electrochemical reactions, the individual concentrations of the active species in the electrolyte are changed. The state of charge of the flow battery is determined by the electrolyte species concentrations, the total flowing electrolyte volume in the system

(tank+pump+hooses+cell), and possibly also by the concentration of solid species on the electrodes. Depending on the cell chemistry the cell can have separated or combined anode and cathode compartments and electrolyte tanks.

Pump

Pb2+

HSO4

Tank

2+PbH+NegativeElectrode(Pb)PositiveElectrode(PbO2/PbO)H

HSO4 +Cell

Figure 1: Working principle of the soluble lead acid flow battery.

1 | SOLUBLE LEAD-ACID REDOX FLOW BATTERY

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In the soluble lead acid flow battery one electrolyte solution is used. The active

component in the electrolyte is the lead ion that reacts on the electrodes to form solid lead (negative electrode) or lead oxide (positive electrode). The electrode chemistry is similar to a traditional lead-acid battery, with the difference that solid lead sulfonate is not formed in the electrodes.

This model simulates a soluble lead-acid flow battery during an applied

charge-discharge load cycle. The surface chemistry of the positive electrode is modeled by using two different lead oxides and two different positive electrode reactions in the model.

Model Definition

CELL GEOMETRY AND MESH

The electrochemical cell consist of two flat 10 cm square electrodes, placed in parallel with a 12 mm gap in between. The aspect ratio of the cell motivates modeling the cell in 2D. The cell geometry and mesh is shown in Figure 2.Outlet

Negative ElectrodePositive Electrode

Inlet

Figure 2: Geometry and mesh of the electrochemical cell.

2 | SOLUBLE LEAD-ACID REDOX FLOW BATTERY

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Due to the very high electrical conductivity of the electrodes, the potential gradients in the electrodes are neglected, and the electrodes are not included in the geometry.To handle possible edge effects in the electrolyte, 1 mm regions are added at the inlet and outlet, outside the active electrode region.

A mapped rectangular mesh is used, and boundary meshing is used to resolve the steep gradients in the electrolyte close to the electrode surfaces.

ELECTROLYTE MASS AND CURRENT TRANSPORT EQUATIONS

The electrolyte is based on a mixture of lead methane sulfonate, methane sulfonic acid and water, which in this model is assumed to dissociate into an electrolyte consisting of Pb2+, H+, HSO4--ions dissolved in a bulk solution of zero-charged species (mainly water). Electroneutrality is assumed locally in the electrolyte. The combination of these assumptions allow for the use of Tertiary Current Distribution, Nernst-Planck interface for modeling the electrolyte transport.

The electric potential in the electrodes is assumed to be space independent. The negative electrode is grounded. On the positive electrode, an electrode potential is calculated in order to fulfill a current density condition defined by the load cycle. (Using the Electrolyte-Electrode Boundary Interface boundary condition).

A load cycle of 1 h charge, 20 s rest, 1 h discharge, 20 s is applied twice to the cell. During charge or discharge a constant current density corresponding to a mean current density in the cell 200 A/m2 is applied.

The species fluxes are defined on the electrode surfaces according to the electrode reactions below. An Inflow condition is used at the inlet with the inlet concentrations (cin,Pb2+ and cin,H+) taken from the tank model described below. An Outflow

condition is set at the outlet. All other boundaries are isolated.

Negative Electrode Reaction

On the negative electrode the following electrode reaction occurs:

Pb2++2e Pb(s)- (1)

with the kinetics being described by a Butler-Volmer expression:

FPbF -----η--- –exp –-----η--- expiPb=Fk0cPb2+ RT RT

Where k0+ is a rate constant and cPb2+ is the concentration of lead ions in the

electrolyte. Pb (2)

3 | SOLUBLE LEAD-ACID REDOX FLOW BATTERY

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As reference electrode we use the negative electrode at reference conditions. The equilibrium potential for the negative electrode is assumed to follow the Nernst equation according to:

RTE0,neg=0V+--------ln(cPb2+)nF (3)

Positive Electrode Main Reaction

The positive electrode main reaction is:

PbO2(s)+4H+2e Pb+-2++2H2O (4)

with the kinetics being described by a Butler-Volmer expression:

iPbO2=Fk0

PbOPbO2cH+FηF----- exp -----η--- –exp –-------- cPb2+---0 RT RT cH+ (5)0where k02 is a rate constant, cH+ is the electrolyte proton concentration and cH+ is the proton reference concentration in the electrolyte at equilibrium.

The positive main reaction has the following equilibrium potential, described by the Nernst Equation:

RT cPb2+ E0,pos=1,8V–--------ln ---------- nF cH+ (6)

Positive Electrode Side Reaction

Multiple types of lead oxides may form on the positive electrode. In this model the following side reaction will be investigated:

PbO2(s)+2H+2e PbO(s)+H2O

where the electrode is kinetics is described by

iPbO=Fk0PbO2 +- (7)FηFηf2bK0cPbOexp -------- –K0cH+cPbO2exp –-------- RT RT (8)

where the overpotential, η, is the same as for the positive electrode main reaction (Equation 6). (The deviation of the equilibrium potential of the side reaction versus the positive main reaction equilibrium potential is controlled by the rate parameters.)

4 | SOLUBLE LEAD-ACID REDOX FLOW BATTERY

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In Equation 8 K0 and K0 are rate constants, and cPbO and cPbO2 are the surface

concentration of the lead oxides (mol/m2).

TANK MODELfb

The electrolyte flowing out from the cell flows into the tank, undergoes mixing, and is then led into the cell again on the inlet side.

Assuming good mixing in the tank the inlet concentrations, cin,Pb2+ and cin,H+, are governed by the following ODEs:

V---d(c2+)=Lin,Pb

V---dc+)=Lin,H outlet(NPb2+ n)dS– inlet(NPb2+ n)dS (9)

outlet(NH+ n)dS– inlet(NH+ n)dS (10)

Where V is the total volume of flowing electrolyte in the tank, and L is the height of the electrodes. (NPb2+ n and NH+ n denote the molar fluxes of the respective

electrolyte species in the normal direction to the boundary).

The two ODEs are modeled using an ODEs and DAEs interface.

FLUID FLOW EQUATIONS

The fluid is led into the cell at a velocity Vin of 2.3 cm/s. The relevant Reynolds number for the flow between the plates is:

ρVinhRe=----------------≈300μ (11)

where the parameter values for water are used for the density ρ, 1000 kg/m3, and viscosity μ, 10-3 Pa·s. We can assume that the flow is in the laminar regime (Re<2000), and hence the Laminar Flow interface is used to model the fluid flow.

Vin is applied at the inlet, a pressure condition is applied to the outlet, and no slip conditions are applied to the electrode surfaces and channel walls. The induced

convection at the electrode surfaces due to the electrochemical reactions is assumed to be negligible. In this way the flow model is stationary and only solved for once. The convective flow is used as a model input to the Tertiary Current Distribution, Nernst-Planck interface.

5 | SOLUBLE LEAD-ACID REDOX FLOW BATTERY