Entropy Increasing Numerical Methods for Prediction of Reversible and Irreversible Heating in Supercapacitors

TL;DR

This work develops entropy-preserving finite-volume schemes for the non-isothermal Poisson–Nernst–Planck–Fourier (PNPF) model of supercapacitors, enabling accurate prediction of reversible and irreversible heating. A first-order semi-implicit Scheme I decouples temperature from ion transport for efficient computation and is proven to preserve mass, positivity, and discrete entropy increase, with unique solvability. A second-order Scheme II uses a modified Crank–Nicolson discretization on the logarithms of concentrations and temperature to maintain positivity and mass conservation while achieving higher accuracy; entropy production is established at the discrete level for the scheme. Numerical experiments demonstrate expected convergence rates, robustness in preserving key physical properties, and the model’s ability to capture temperature oscillations and a quadratic scaling of heating with voltage-scan rate, underscoring the impact of complex electrode geometry on heat generation dynamics. These results provide a reliable tool for predicting non-isothermal electrokinetics and heating in porous-electrode supercapacitors, with implications for design and thermal management.

Abstract

Accurate characterization of entropy plays a pivotal role in capturing reversible and irreversible heating in supercapacitors during charging/discharging cycles. However, numerical methods that can faithfully capture entropy variation in supercapacitors are still in lack. This work develops first-order and second-order finite-volume schemes for the prediction of non-isothermal electrokinetics in supercapacitors. Semi-implicit discretization that decouples temperature from ionic concentrations and electric potential results in an efficient first-order accurate scheme. Its numerical analysis theoretically establishes the unique solvability of the nonlinear scheme with the existence of positive ionic concentrations and temperature at discrete level. To obtain an entropy-increasing second-order scheme, a modified Crank-Nicolson approach is proposed for discretization of the logarithm of both temperature and ionic concentrations, which is employed to enforce numerical positivity. Moreover, numerical analysis rigorously demonstrates that both first-order and second-order schemes are able to unconditionally preserve ionic mass conservation and original entropy increase for a closed, thermally insulated supercapacitor. Extensive numerical simulations show that the proposed schemes have expected accuracy and robust performance in preserving the desired properties. Temperature oscillation in the charging/discharging processes is successfully predicted, unraveling a quadratic scaling law of temperature rising slope against voltage scanning rate. It is also demonstrated that the variation of ionic entropy contribution, which is the underlying mechanism responsible for reversible heating, is faithfully captured. Our work provides a promising tool in predicting reversible and irreversible heating in supercapacitors.

Figures (7)

• Figure 1: Delaunay mesh with blue solid vertices ($\hbox{\boldmath$x$}_i$, $\hbox{\boldmath$x$}_j, \cdots$) and dual Voronoi control volumes ($V_i$, $V_j, \cdots$) with blue solid edges. The $\hbox{\boldmath$n$}_{i,\sigma}$ is the unit outward vector normal to the edge $\sigma$ of $V_i$.
• Figure 2: Numerical error of $c^1$, $c^2$, $\psi$, and $T$ at time $T=0.1$ obtained by (a) Scheme I with a mesh ratio $\Delta t=h^2$ and (b) Scheme II with a mesh ratio $\Delta t=h/10$.
• Figure 3: A schematic illustration of the computational domain $\Omega$ filled with electrolytes shown in gray (upper) and the mesh (lower) contained within a rectangle $[-10,10]\times[0,10] ~{\rm nm}^2$. Biased voltage differences are applied across two thermally-insulated, blocking electrodes with complex interfaces denoted by $\Gamma_2$ and $\Gamma_3$. The boundary in the middle is labeled by $\Gamma_1$.
• Figure 4: Evolution of the discrete entropy $S$ (rescaled by $S_0:=k_Bc_0L^2$) and total mass of cations (Left), as well as minimum concentration and temperature over the computational mesh (Right).
• Figure 5: Evolution of the cation concentration $c^1$, the electric potential $\psi$, and temperature rise $\Delta T:=T-T_0$ at time $t=0$$t=0.1$, $t=1$ and $t=30 ({\rm \mu s})$.

Theorems & Definitions (9)

• Theorem 2.1
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6
• Remark 3.7
• Remark 3.8