Flow Between

Flow between two or more compartments follow the mechanics described below. In the below figure, we have flow between two compartments. We have species \(A_1\) flowing out of compartment 1 into compartment 2 as \(A_2\).

The flows from a compartment are derived as the flow rate (F) multiplied by the concentration of the species leaving the compartment as a minus term. Flow to a compartment is the same derivation without the minus term. The flows in the above diagram would be derived as:

\[\begin{split}\begin{align*} V_1 * \frac{d[A_1]}{dt} &= -F * A_{1} \\ V_2 * \frac{d[A_2]}{dt} &= F * A_{1} \end{align*}\end{split}\]

Split Flow

Often we need to split the flow from one compartment to multiple compartments. This option splits the flow from one compartment to go to multiple compartments. In the figure below, we see the flow from compartment 1 is split to go to compartment 2 and compartment 3.

The summation of the output flows will be equal to the input flow:

\[F_{out} = \sum_{1}^{n} F_{in}\]

Given the above figure this equation derives out to:

\[F = F_1 + F_2\]

The resulting flow differential equations derive as:

\[\begin{split}\begin{align*} V_{1} \frac{dA_{1}}{dt} = -F * A_{1} \\ V_{2} \frac{dA_{2}}{dt} = -F_{1} * A_{1} \\ V_{3} \frac{dA_{3}}{dt} = -F_{2} * A_{1} \\ \end{align*}\end{split}\]