(1)	$\begin{array}{l}\displaystyle \dot m = \sum_\alpha \dot m_\alpha\end{array}$

(2)	$\begin{array}{l}\displaystyle A = \sum_\alpha A_\alpha\end{array}$

(3)	$\begin{array}{l}\displaystyle \sum_\alpha s_\alpha = 1\end{array}$

(4)	$\begin{array}{l}\displaystyle u_m = \sum_\alpha s_\alpha \cdot \dot u_\alpha\end{array}$

(5)	$\begin{array}{l}\displaystyle q_\alpha = \dot m_\alpha / \rho_\alpha = A_\alpha \, u_\alpha \Rightarrow \dot m_\alpha = \rho_alpha \, A_\alpha \, u_\alpha\end{array}$

(6)	$\begin{array}{l}\displaystyle s_\alpha = \frac{\dot m_\alpha}{\rho_\alpha \, u_\alpha} \cdot \left( \sum_\beta \frac{\dot m_\beta}{\rho_\beta \, u_\beta} \right)^{-1}\end{array}$

For homogeneous pipe flow: $\begin{array}{l}u_\alpha = u_m, \, \forall \alpha\end{array}$ and volumetric shares are going to be:

(7)	$\begin{array}{l}\displaystyle s_\alpha = \frac{\dot m_\alpha}{\rho_\alpha} \cdot \left( \sum_\beta \frac{\dot m_\beta}{\rho_\beta} \right)^{-1}\end{array}$

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See also