Table 2 Governing equations in the shell of GLMC26,27,28,29,30.
Mass transfer | Momentum transfer |
|---|---|
\(\frac{{\partial C_{i} }}{\partial t} = - \nabla N_{i} + R_{i} \;\;\;\;(1)\) | \(\begin{aligned} V_{z,s} & = 2\overline{{V_{s} }} \left[ {1 - \left( {\frac{{r_{2} }}{{r_{3} }}} \right)^{2} } \right] \\ & \quad \times \left[ {\frac{{(r/r_{3} )^{2} - (r_{2} /r_{3} )^{2} + 2\ln \left( {r_{2} /r} \right)}}{{3 + (r_{2} /r_{3} )^{4} - 4(r_{2} /r_{3} )^{2} + 4\ln \left( {r_{2} /r_{3} } \right)}}} \right]\;\;\;\;(4) \\ \end{aligned}\) |
\(N_{i} = - D_{i} \nabla C_{i} + C_{i} V_{Z } \;\;\;\;\left( 2 \right)\) | \(1 - \varphi = \frac{{nr_{2}^{2} }}{{R^{2} }}\;\;\;\;\left( 5 \right)\) |
\(\begin{aligned} & D_{{CO_{2} ,t}} \left[ {\frac{{\partial^{2} C_{{CO_{2} ,t}} }}{{\partial r^{2} }} + \frac{1}{r}\frac{{\partial C_{{CO_{2} ,t}} }}{\partial r} + \frac{{\partial^{2} C_{{CO_{2} ,t}} }}{{\partial z^{2} }}} \right] \\ & \quad = V_{z,t} \frac{{\partial C_{{CO_{2} ,t}} }}{\partial z}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;(3) \\ \end{aligned}\) | \(r_{3} = r_{2} \sqrt {1/\left( {1 - \varphi } \right)} \;\;\;\;(6)\) |
