Convective mass transfer for turbulent flow (acidizing)

Input(s)

$D_{A}$: Diffusion Coefficient of Component A $\left(\mathrm{cm}^{2} / \mathrm{s}\right)$

$D_{E}$ : Effective Diffusion Coefficient of Component A $\left(\mathrm{cm}^{2} / \mathrm{s}\right)$

$d c_{a} / d Y$ : Concentration Gradient $\left(\mathrm{mol} / \mathrm{cm}^{4}\right)$

$c_{a}$ : Concentration of Flowing Acid $\left(\mathrm{mol} / \mathrm{cm}^{3}\right)$

$<c_{a}>$ : Average Concentration of Flowing Acid $\left(\mathrm{mol} / \mathrm{cm}^{3}\right)$

$<c_{a}(w)>$ : Average Concentration of Flowing Acid $\left(\mathrm{mol} / \mathrm{cm}^{3}\right)$

$V_{N}$ : Fluid Velocity Normal to the Surface $(\mathrm{cm} / \mathrm{s})$

$\left\langle V_{N}\right\rangle$ : Average Fluid Velocity Normal to the Surface $(\mathrm{cm} / \mathrm{s})$

$K_{g}$ : Effective Mass Transfer Coefficient $(\mathrm{cm} / \mathrm{s})$

Output(s)

$U_{a, y}$: Diffusion Flux of a Component A in $\mathrm{Y}$ direction $\left(\mathrm{mol} / \mathrm{cm}^{2} \mathrm{~s}\right)$

$\left\langle U_{a, y}\right\rangle$ : Average Diffusion Flux of a Component A in Y direction $\left(\mathrm{mol} / \mathrm{cm}^{2} \mathrm{~s}\right)$

Formula(s)

$$ \begin{gathered} <U_{a, y}>=-D_{A} \frac{d<c_{a}>}{\partial Y}+<c_{a} V_{N}> \\ <U_{a, y}>=-D_{E} \frac{d<c_{a}>}{d Y}+<c_{a}><V_{N}> \\ <U_{a, y}>=K_{g}\left[<c_{a}>-<c_{a}(w)>\right]+<c_{a}><V_{N}> \end{gathered} $$

Reference(s)

Williams, B. B., Gidley, J. L., \& Schechter, R. S. (1979). Acidizing fundamentals. Henry L. Doherty Memorial Fund of AIME, Society of Petroleum Engineers of AIME, Page 23.