You are viewing an old version of this page. View the current version.
Compare with Current
View Page History
« Previous
Version 18
Next »
| (1) |
\frac{\partial (\rho_A \, \phi) }{\partial t} + \nabla (\rho_A \, {\bf u}_A) = 0, \quad A = \{ O, G, W \} |
|
| (2) |
\int_V \, \frac{\partial (\rho_A \, \phi) }{\partial t} \, dV = - \int_V \, \nabla (\rho_A \, {\bf u}_A) \, dV = - \int_{\partial V} \, \rho \, {\bf u}_A \, d {\bf A} |
|
| (3) |
V \cdot \delta (\rho_A \, \phi) = \delta \, m_A \Rightarrow V \cdot \delta \left( \phi \, \sum_\alpha \rho_{A,\alpha} \, s_\alpha \right) = \mathring{\rho}_A \cdot \delta \, q_A, \quad \alpha = \{ o, g, w \} |
|
| (4) |
\Rightarrow \delta \left( \phi \, \sum_\alpha \frac{\rho_{A,\alpha}}{\mathring{\rho}_A} \, s_\alpha \right) = V^{-1} \cdot \delta \, q_A \Rightarrow \delta \left( \phi \, \sum_\alpha \frac{\mathring{V}_{A,\alpha}}{V_\alpha} \, s_\alpha \right) = V^{-1} \cdot \delta \, q_A |
|
| (5) |
\delta \left( \phi \, \sum_\alpha \frac{\mathring{V}_{A,\alpha}}{V_\alpha} \, s_\alpha \right) = V^{-1} \cdot \delta \, q_A |
|
For the MBO fluid:
| (6) |
\xi_{A, \alpha} = \frac{\mathring{V}_{A,\alpha}}{V_\alpha} |
|
| (7) |
\xi_{O, o} = \frac{\mathring{V}_{O, o}}{V_o} = \frac{1}{B_o} |
|
| (8) |
\xi_{O, g} = \frac{\mathring{V}_{O, g}}{V_g} =
\frac{\mathring{V}_{O, g}}{\mathring{V}_{G, g}} \cdot \frac{\mathring{V}_{G, g}}{V_g}
= \frac{R_s }{B_g} |
|
| (9) |
\xi_{G, o} = \frac{\mathring{V}_{G, o}}{V_o} = \frac{\mathring{V}_{G, o}}{\mathring{V}_{O, o}} \cdot \frac{\mathring{V}_{O, o}}{V_o} =\frac{R_s }{B_o} |
|
| (10) |
\xi_{G, g} = \frac{\mathring{V}_{G, g}}{V_g} = \frac{1}{B_g} |
|
| (11) |
\xi_{W, w} = \frac{\mathring{V}_{W, w}}{V_w} = \frac{1}{B_w} |
|
|
|
|
Next step is to write the equations explicitly for MBO fluid.
| (12) |
\delta \left[ \phi \cdot \left( \xi_{O,o} \, s_o + \xi_{O,g} \, s_g \right) \right] = V^{-1} \, \delta q_O |
|
| (13) |
\delta \left[ \phi \cdot \left( \xi_{G,o} \, s_o + \xi_{G,g} \, s_g \right) \right] = V^{-1} \, \delta q_G |
|
| (14) |
\delta \left[ \phi \cdot \xi_{Ww} \, s_w \right] = V^{-1} \, \delta q_W |
|
|
