Mixing and Compartment Models

Definition Let $M(t)$ be the amount of a substance in a compartment at time $t$. A mass balance states that

where $r_{\text{in}}$ and $r_{\text{out}}$ are the rates at which the substance enters and leaves. If the substance is also produced or used up inside the compartment, for example by a reaction or by decay, that rate is added with its sign. Every term has units of amount per time. A stream with volume flow rate $q$ and concentration $c$ carries the substance at the rate $qc$, so a single inlet contributes $r_{\text{in}} = q_{\text{in}}c_{\text{in}}$.

Definition The amount $M(t)$ is the quantity of the substance in the compartment, for example grams of salt. The concentration is the amount per unit volume of liquid,

defined while the volume $V(t)$ is positive, and measured in grams per liter when $M$ is in grams and $V$ in liters. Balances are written for the amount, because amounts add up and are what the streams carry in and out. When the volume changes, the quotient rule gives

so $c'$ differs from $M'/V$ by the term $-MV'/V^2$, and the amount and the concentration can even change in opposite directions.

Definition A compartment is perfectly mixed, or well mixed, if the substance is spread uniformly through it at every instant. Then the liquid leaving through an outlet has the current concentration $M(t)/V(t)$, and the outflow rate in the mass balance is

With this rule the balance becomes one equation for the single unknown $M$. It is a modeling assumption and not a physical law. Layers of different concentration, slow stirring or a delay between inlet and outlet make the outlet concentration differ from $M/V$, and then the equation describes a different tank.

Definition The liquid volume obeys its own balance $V' = q_{\text{in}} - q_{\text{out}}$. With constant flows

and the model applies only while $V(t)$ stays positive and within the capacity of the tank. With perfect mixing, constant flows and a constant incoming concentration, the amount satisfies the linear first-order equation

Its coefficient $q_{\text{out}}/V(t)$ depends on time whenever the volume changes and liquid drains, and the equation is solved with the integrating factor $\mu(t) = \exp\bigl(\int q_{\text{out}}/V(t)\,dt\bigr)$. Since $M = cV$ and $V' = q_{\text{in}} - q_{\text{out}}$, the same assumptions give an equation for the concentration,

So while liquid flows in, the concentration moves toward $c_{\text{in}}$ without ever crossing it, and it stays at $c_{\text{in}}$ if it starts there. While the tank only drains, the concentration stays constant.

Definition A compartment model splits a system into well-mixed compartments with amounts $M_1, \dots, M_n$ and volumes $V_1, \dots, V_n$, and writes one mass balance for each. A flow $q$ from compartment $i$ to compartment $j$ removes the substance from compartment $i$ at the rate $qM_i/V_i$ and adds it to compartment $j$ at the same rate. Each volume follows its own balance and stays constant when as much liquid flows into the compartment as flows out. The balances form a system of first-order equations, coupled because the rate of change of each amount depends on the others. In a closed system, where no substance enters or leaves from outside and none is produced or used up, each transfer term appears once with each sign. The balances then add up to zero, and the total amount $M_1 + \cdots + M_n$ is constant.