C.29.BB:4.3 - Combine the local balances

Write each included part’s change using the selected terms. Sum those equations. Matching internal transfers cancel, leaving

Q(S, t1) - Q(S, t0)
    = total inward transfer - total outward transfer
      + internal production - internal removal.

The transfer terms are amounts over the interval. A rate must be integrated over that interval, or multiplied by its duration when the rate is constant. The signs follow the chosen boundary.

For a differentiable rate model, the corresponding instantaneous relation is

dQ(S,t)/dt = inward rate - outward rate + production rate - removal rate.

Use the finite-interval form for discrete events or when a rate description adds no help. If the selected boundary moves, count crossings relative to that moving boundary using the appropriate transport relation.

A closed boundary eliminates crossing terms only when the model excludes those exchanges. Constancy of Q additionally requires net internal production and removal to cancel. Preserve that premise in any conservation conclusion.