Vanadium Flow Battery: Pumping and Shunt Current Loss Optimiser

The electrical model follows the equivalent-circuit approach of Xing, Zhang and Ma (2011), with the four-rail topology checked against Glazkov et al. (2022), Section 2.1. Positive and negative electrolytes each have an inlet and outlet manifold. Identical inlet/outlet rails are reduced by symmetry; positive taps connect at cell node k and negative taps at node k−1. Every cell retains its resistance and EMF.

For a circular passage, $R_c=L_c/(\sigma A_c)$ and each manifold segment has $R_m=p/(\sigma A_m)$. Kirchhoff equations for the 2(N−1) manifold mesh currents form a symmetric pentadiagonal system. The solution gives each channel current, manifold current and cell reaction current. Shunt loss is the sum of $I^2R$ over all four rails, not stack current multiplied by channel resistance.

Hydraulics uses Darcy flow through the compressed felt, $\Delta p_f=\mu L_f q/(K A_f)$, and $K=d_f^2\epsilon^3/[K_{KC}(1-\epsilon)^2]$. The coefficient convention is explicit. The example uses an assumed effective coefficient of 24 in this convention. Xing et al. (2015) reports measured calibration of a felt coefficient; that paper's numerical coefficient must not be transferred without checking its geometric convention. Circular channels, pipes and each manifold segment use Darcy–Weisbach with Churchill's (1977) continuous friction correlation, converted to the Darcy convention, plus the entered minor-loss coefficients. Segment flow decreases by one cell flow at each tap. Both-loop pump input is $2Q\Delta p_{eq}/\eta_p$, where manifold dissipation is weighted by each segment's actual flow.

Flow factor is $Q/[NI/(Fc x_{react})]$. Uniform reaction along each flow-through felt gives $c_{react}(z)=c x_{react}-|I_k|z/(FqL_f)$. The Tang velocity law is $k_m=k_{ref}(u_p/1\,\mathrm{m/s})^b$, with $u_p=q/(\epsilon A_f)$; defaults are $k_{ref}=1.6\times10^{-4}$ m/s and $b=0.4$. Each half-cell has $j_{lim}=Fk_m c_{react}$ and film concentration overpotential $-(RT/F)\ln(1-|j|/j_{lim})$. Current density uses projected active area. The two half-cell losses are added to the local ideal bulk Nernst polarisation.

The logarithmic voltage expression is integrated analytically over the felt length, retaining the velocity-dependent film loss and changing bulk concentration. The resulting cell EMFs and the exact shunt circuit are iterated to 10⁻⁹ V, up to 12 passes. Cell reaction current, including shunts, sets both bulk depletion and electrode-surface depletion. Outlet limiting-current fraction and surface concentration are checked for every cell. Invalid trial voltages are regularised only to keep the solver finite; such trials cannot be accepted by the search.

The constrained search varies flow factor, feed diameter and manifold diameter. For discharge pumping-plus-shunt optimisation, a Newton solve for the limiting-current flow bound is coupled to the circuit; the bulk-utilisation bound is also enforced. Other objectives sample flow as well as both diameters. A seed grid is followed by shrinking neighbouring searches. Every candidate must satisfy the surface-transport, bulk-utilisation, header-uniformity and felt Reynolds-number limits. The recommended candidate is checked again.

Chemical conversion at the inlet state is balanced against terminal power, cell resistance, bulk concentration loss, surface concentration loss and ionic Joule loss. Instantaneous discharge efficiency subtracts both pumps from terminal output and divides by chemical conversion power; charge efficiency divides stored chemical power by terminal demand plus both pumps.

A single series stack has parallel, identical flow-through felt half-cells and circular feed passages. Both electrolytes have the same viscosity, density, geometry, flow, pump efficiency and mass-transfer law, but may have different conductivities. Inlet and outlet ionic paths on a given side are identical. Properties are constant at the entered temperature and SOC; temperature changes Nernst and film voltages, not the entered material properties.

The closed tanks provide no external electrical shunt or net static lift. Branch flows are assumed equal; manifold friction is summed segment by segment. Ideal activities, balanced half-cell reactant fractions and uniform projected reaction current along the felt are assumed. The empirical mass-transfer coefficient uses the lumped projected-area convention of the requested Tang treatment; it must be calibrated in that same convention.

Default properties and dimensions are illustrative inputs, not a product specification. The schematic shows representative cells and one electrolyte loop; both loops and every entered cell are calculated. The model does not reproduce Tang's full charge–discharge cycle simulations.

A design is admissible only when every cell has positive voltage and the intended current direction, bulk utilisation is below its entered limit, and outlet |j|/(F k_m c) is below the entered limiting-current fraction with positive surface reactant concentration. The inlet supply check alone is insufficient. The fibre Reynolds number, based on superficial velocity and fibre diameter, must be at most one for every search candidate and the final recommendation. Darcy flow neglects inertial drag and anisotropy; measure permeability for the actual felt compression and flow direction.

The U-header pressure spread must be at most 10% of the felt-plus-feed drop. This screens the equal-flow assumption; pressure drop is an energy-equivalent value, not a pump shut-off head or a detailed branch-pressure solution. Junction momentum recovery is omitted; include measured junction losses in the minor-loss budget when material. If the lower flow endpoint is infeasible or no positive feasible interval remains, the flow charts contain no samples and an input message explains what to change.

The search gives a bounded numerical candidate, not a proof of the global minimum. Through 60 cells it uses a 5 × 5 diameter seed grid and five neighbourhood refinements for discharge parasitic loss. Other objectives use a 3 × 3 × 5 diameter/flow seed grid and three refinements. Above 60 cells, diameter grids are 3 × 3 with two refinements; above 90 cells one axis-wise refinement is used. Reduced search resolution keeps the full coupled physics within a few seconds, but can miss narrow feasible regions or secondary minima. Narrow the bounds around a promising result to refine it.

Curves are solved at 41 flow and 31 diameter points through 60 cells; at 31 and 17 points through 90 cells; and at 21 and 9 points above 90 cells. The pictures use PCHIP interpolation to 81 flow and 61 diameter points. This reduces runtime while retaining every circuit cell and the complete transport model, but can smooth narrow features. The diameter comparison is unconstrained and may extend beyond accepted transport, header or utilisation limits.

The default mass-transfer correlation is empirical, not universal for every felt or compression. Fit its coefficient and exponent to your electrode using the stated pore-velocity and projected-area conventions; changing the area convention changes the required coefficient. Tai et al.'s Table 3 provides independent checks of this transport branch. Its rounded limiting currents differ slightly from F k c, which is retained without fitting.

Crossover, nonideal activities, gas generation, tank dynamics, ageing, thermal feedback and full-cycle voltage cut-offs are omitted. Instantaneous efficiency is not round-trip efficiency. Tang et al. appeared online in 2013 and in Journal of Power Sources 248 (2014); their cycle-optimal factor 7.5 is not a universal target.