2.1: Compartment Diagrams & Balance Law

• Many processes may be considered as compartment models.
• This is the case when the process has inputs and outputs from a compartment over time.

• The compartment is the atmosphere
• Input comes from many processes, such as burning, etc
• Output occurs through plant respiration, etc

• Model amount of \( \mathrm{CO}_2 \) in atmosphere at any given time.
  
• Input of appears to exceed output, hence an increase of \( \mathrm{CO}_2 \) in compartment.

• Balance law for \( \mathrm{CO}_2 \) in compartment:

\[ \small{ \mathrm{amount ~CO_2} = \mathrm{ output~ amount } - \mathrm{ input~ amount } } \]

• Alternatively, we can write this as

\[ \small{ \left\{\mathrm{CO}_2 \right\} = \left\{\mathrm{CO}_2\right\}_o - \left\{\mathrm{CO}_2\right\}_i } \]

• In terms of rates over a time interval of length \( \Delta t \):

\[ \small{ \left\{\frac{\Delta \mathrm{CO}_2}{\Delta t} \right\} = \left\{\frac{\Delta\mathrm{CO}_2}{\Delta t}\right\}_o - \left\{\frac{\Delta\mathrm{CO}_2}{\Delta t} \right\}_i } \]

• Balance law in terms of rates over \( \Delta t \):

\[ \small{ \begin{aligned} \left\{\frac{\Delta \mathrm{CO}_2}{\Delta t} \right\} = \left\{\frac{\Delta\mathrm{CO}_2}{\Delta t}\right\}_o - \left\{\frac{\Delta\mathrm{CO}_2}{\Delta t} \right\}_i \end{aligned} } \]

• In words:

\[ \small{ \begin{aligned} \left\{\mathrm{net~rate~of~change} \right\} = \left\{\mathrm{rate~out}\right\} - \left\{\mathrm{rate~in}\right\} \end{aligned} } \]

• Word equation for balance law for a substance:

\[ \small{ \begin{Bmatrix} \mathrm{net \, rate \, of \, change} \\ \mathrm{of \, a \, substance} \end{Bmatrix} = \begin{Bmatrix} \mathrm{rate~out} \end{Bmatrix} - \begin{Bmatrix} \mathrm{rate~in} \end{Bmatrix} } \]

• The balance law provides a way of relating variables in such a way that a differential equation model can be developed.

• Decay of radioactive elements
• Births and deaths in a population
• Pollution into and out of lake, river, or atmoshere
• Drug assimilation into, and removal from, the bloodstream