q_O =  \frac{q_o}{B_o} + \frac{R_v \, q_g}{B_g}

q_G =\frac{q_g}{B_g} + \frac{R_s \, q_o}{B_o}

q_W =  \frac{q_w}{B_w}

q_o = \frac{ B_o \cdot ( q_O - R_v \, q_G) }{1- R_v \, R_s}

q_g = \frac{ B_g \cdot ( q_G - R_s \, q_O)}{1- R_v \, R_s}

q_w = B_w \cdot q_W

\rho_o = \frac{
\dot m_o}{q_o}= \frac{\rho_O + \rho_G \, R_s}{B_o}

\rho_g = \frac{\dot m_g}{q_g}= \frac{\rho_G + \rho_O \, R_v}{B_g}

\rho_w =\frac{\dot m_w}{q_w}=  \frac{\rho_W}{B_w}

\dot m_O = \rho_O \cdot q_O

\dot m_G = \rho_G \cdot q_G

\dot m_W = \rho_W \cdot q_W

\dot m_o = \rho_o \cdot q_o = (\rho_O + \rho_G \, R_s) \cdot \frac{ q_o}{B_o}

\dot m_g = \rho_g \cdot q_g = (\rho_G + \rho_O \, R_v) \cdot \frac{q_g }{B_g}

\dot m_w = \rho_w \cdot q_w

\dot m_o = 
\frac{(\rho_O + \rho_G \, R_s) \cdot ( q_O - R_v \, q_G) }{1- R_v \, R_s}

\dot m_g = 
\frac{ (\rho_G + \rho_O \, R_v) \cdot ( q_O - R_v \, q_G) }{1- R_v \, R_s}

\dot m_w = \rho_W \cdot \frac{q_w}{B_w} = \rho_W \cdot q_W

\dot m = \dot m_o + \dot m_g + \dot m_w = \dot m_O + \dot m_G + \dot m_G 

q_t = q_o + q_g + q_w

q_t = \frac{B_o - B_g \, R_s}{1-R_v \, R_s} \cdot q_O 

+\frac{B_g - B_o \, R_v}{1-R_v \, R_s} \cdot q_G 

+ B_w \cdot q_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 }

\rho = s_o \, \rho_o + s_g \, \rho_g + s_w \, \rho_w

c = s_o \, c_o + s_g \, c_g + s_w \, c_w