Boundary Conditions

Inlet ($\Gamma_{\text{inlet}}$)

A uniform volume-averaged velocity is set:

$$\vec{v}_{\text{inlet}} = -\frac{Q}{A_{\text{inlet}}}\vec{n}$$

where $\vec{n}$ is the outward unit normal vector. For IDFF, the total flow rate $Q$ is divided equally among the multiple inlet channels. Pressure is set to zero-gradient, and species concentrations are prescribed as inlet values $C_{\text{O,in}}$ and $C_{\text{R,in}}$.

Outlet ($\Gamma_{\text{outlet}}$)

• Velocity: zero-gradient
• Pressure: uniform reference $p = 0$

The overall pressure drop $\Delta p$ is obtained from simulations by averaging the pressure over the inlet boundary:

$$\Delta p = \frac{\int_{\Gamma_{\text{inlet}}} p \, d\Gamma}{\int_{\Gamma_{\text{inlet}}} d\Gamma}$$

Current Collector ($\Gamma_{\text{cc}}$)

• $\Phi_{\text{s}} = 0$ V (reference potential)
• $\Phi_{\text{l}}$: zero-gradient (no ionic current on bipolar plate)

Membrane ($\Gamma_{\text{mem}}$)

• $\Phi_{\text{s}}$: zero-gradient (no electronic current through membrane)
• $\Phi_{\text{l}}$: depends on the operation mode (see below)

Potentiostatic mode

A fixed, homogeneous value of $\Phi_{\text{l}}$ is set:

$$\Phi_{\text{l,mem}} = -(U_0 + \eta_{\text{HC,sim}})$$

The average current density is computed as:

$$i_{\text{avg,sim}} = \frac{\int_{\Gamma_{\text{mem}}} \vec{i_{\text{l}}} \cdot \vec{n} \, d\Gamma}{\int_{\Gamma_{\text{mem}}} d\Gamma}$$

Galvanostatic mode

A Neumann condition enforces a homogeneous applied ionic current density:

$$i_{\text{app,mem}} = -\kappa^{\text{eff}} \frac{\partial\Phi_{\text{l}}}{\partial\vec{n}}$$

The half-cell overpotential is then computed from the simulated fields.

Walls ($\Gamma_{\text{walls}}$)

• Velocity: no-slip
• Pressure: zero-gradient
• All other fields: zero-gradient

Summary Table

Symbol	$\Gamma_{\text{inlet}}$	$\Gamma_{\text{outlet}}$	$\Gamma_{\text{mem}}$	$\Gamma_{\text{cc}}$	$\Gamma_{\text{walls}}$
$\vec{v}$	Fixed velocity	Zero-gradient	No-slip	No-slip	No-slip
$p$	Zero-gradient	$p = 0$	Zero-gradient	Zero-gradient	Zero-gradient
$C_{\text{O}}$	$C_{\text{O,in}}$	Zero-gradient	Zero-gradient	Zero-gradient	Zero-gradient
$C_{\text{R}}$	$C_{\text{R,in}}$	Zero-gradient	Zero-gradient	Zero-gradient	Zero-gradient
$\Phi_{\text{s}}$	Zero-gradient	Zero-gradient	Zero-gradient	$\Phi_{\text{s}} = 0$	Zero-gradient
$\Phi_{\text{l}}$	Zero-gradient	Zero-gradient	Fixed / Flux	Zero-gradient	Zero-gradient

Half-Cell Overpotential

The total simulated potential loss in the half-cell:

$$\eta_{\text{HC,sim}} = \frac{\int_{\Gamma_{\text{mem}}}(-\Phi_{\text{l}} - U_0) \, \vec{i_{\text{l}}} \cdot \vec{n} \, d\Gamma}{\int_{\Gamma_{\text{mem}}} \vec{i_{\text{l}}} \cdot \vec{n} \, d\Gamma}$$

In either operational mode, the pair $(\eta_{\text{HC,sim}}, \, i_{\text{avg,sim}})$ provides a data point for simulated polarization curves.