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\dot m_w = \rho_W \cdot q_w/B_w

\dot m = \dot m_o_O + \dot m_gG + \dot m_w W 

\rhodot m_w = \rho_W \cdot q_w/B_w

q_t = \frac{B_o - B_g \, R_s}{1-R_v \, R_s } \cdot \frac{\dot m_O }{\rho_O}

+\frac{B_g - B_o \, R_v}{1-R_v \, R_s } \cdot \frac{\dot m_G }{\rho_G}

+ B_w\cdot \frac{\dot m_W}{\rho_W}

\rho = \frac{\dot m}{q_t} = \frac{\dot m_O + \dot m_G + \dot m_G}{
\frac{B_o - B_g \, R_s}{1-R_v \, R_s } \cdot \frac{\dot m_O }{\rho_O}

+\frac{B_g - B_o \, R_v}{1-R_v \, R_s } \cdot \frac{\dot m_G }{\rho_G}

+ B_w\cdot \frac{\dot m_W}{\rho_W}
}

s_o = \frac{q_o}{q_t} = \frac{ B_o \, (q_O - R_v \, q_G)}{(B_o - B_g \, R_s) \, q_O + (Bg - B_o \, R_v) \, q_G + B_w \, (1- R_v \, R_s) \, q_W }

s_g = \frac{q_g}{q_t} = \frac{ B_g \, (q_G - R_s \, q_O)}{(B_o - B_g \, R_s) \, q_O + (Bg - B_o \, R_v) \, q_G + B_w \, (1- R_v \, R_s) \, q_W }

s_w = \frac{q_w}{q_t} = \frac{ B_w \, (1- R_v \, R_s) \, q_W}{(B_o - B_g \, R_s) \, q_O + (Bg - B_o \, R_v) \, q_G + B_w \, (1- R_v \, R_s) \, q_W }