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Section 2.4 Vanadium crossover Many other system level studies only consider the effect of diffusion, which is modeled using Fick’s law [16, 19, 27–31]. The presented models assume that no actual mass transfer across the membrane takes place which is also assumed in this work. 2.4.1 Vanadium crossover flux Although the total vanadium concentration is the same in the negative and positive electrolytes, a concentration gradient exists because the vanadium ions exist in different oxidation states on the positive and negative half-sides. Driven by this concentration gradient, vanadium ions may cross the membrane and diffuse into the opposite half-cell. This diffusion can be described using the diffusion flux density, ji, of the vanadium species i, described by Fick’s first law, shown in Eq. (2-14) [28, 32]. The diffusion flux density depends on the gradient of the concentration ci and the diffusion coefficient Di. The diffusion coefficient is a measure for the membrane’s resistivity against the diffusion of vanadium ions. j D dci , i∈{2,3,4,5} i iMem dx Wherein: ci Concentration of vanadium species i DiMem Diffusion coefficient of vanadium species i in the membrane ji Ionic flux density of vanadium species i through the membrane x Location (2-14) (molm-3) (m2s-1) (mols-1m-2) (m) In this work, no spatial distribution of any process is considered. Within each half-cell, perfect mixture is assumed to take place. Thus, the concentration gradient is the difference in the respective species’ concentration on both half-sides, divided by the membrane thickness, δMem. Consequently, we calculate the effective ionic flux, JiC, by multiplying the flux density with the membrane area, AMem. The membrane area corresponds to the electrode area, AE, as shown in Eq. (2-15) for the negative half-cell and in Eq. (2-16) for the positive half-cell. J iC D D iMem ciC ciC A Mem Mem ciC ciC A Mem Mem , i∈{2,3,4,5} , i∈{2,3,4,5} (2-15) (2-16) iMem 2.4.2 Self-discharging reactions J iC If vanadium ions enter the respective opposite half-cell, they partake in redox reactions as long as the required reaction partners are present [32, 33]. We can express these reactions as follows. 20PDF Image | Model-based Design Vanadium Redox Flow Batteries
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