First lets go through the mathematical terms we intend to use
$$ Reservoir - Q $$
$$Time - t $$
A reservoir is the "box" from the one box model, where the substances are accumulated. It is denoted by the letter Q which suggest that it is an expression of the quantity of substance in the reservoir.
Time denotes the times since the reservoir started filling and is denoted by the letter t
$$Flux - \nu $$
These terms are demonstrated in the following Lab-Cast
This model forecast
Reservoir size is equal to the time since the flux began doubled by the flux
$$ Q={\nu}*{t} $$
The flux is equal to the size of the reservoir devided by the time since the flux began
$$\nu = \frac{Q}{t}$$
The time since the flux began is equal to the reservoir size devided by the flux
$$t=\frac{Q}{\nu}$$
Naturlly, fluxes can come in many different shapes and can be combined or detached. For instance, lets add another flux of dissolved tracer to our water flux.
Nothing changes in our mathematics the only thing we need to adjust is that we need for each tracer using the appropriate terms. The link between the water and the tracer flux is the Concentration in the flux and in the reservoir.
reservoir concentration - $C_r$ and flux concentration - $C_f$
So the terms used for tracer calculation (marked as tr):
$$Q_{tr}=t*\nu*C_f = C_r*Q$$
$$\nu_{tr}=\nu_{tr}*C_{f}$$
$$t=\frac{Q_{tr}}{\nu_{tr}}$$
The most important statement in this post is:
In the real world every flux has an opposing flux (outflux), otherwize, Reservoir size (Q) will reach infinity.
So lets introduce a new term:
$$Removal - R$$
In the following lab cast an example of influx and removal is given:
And now for some math, this time using the two fluxes situation.
Let us try and calculate the change of tracer concentration $Q_{tr}$ per unit of time:
$$\frac{dQ_{tr}}{dt}= \Sigma sources - \Sigma sinks=f_{tr}-R_{tr}$$
In our simple example the only source is the particulate source through the tap and the only sink is the particles settling. The size of the sink is linearly related to the size of the tracer reservoir $Q_{tr}$. Let us define the linear relationas k.
$$\frac{dQ}{dt}=f_{tr}-kQ_{tr}$$ after doing some algebra: $$\frac{dQ}{f_{tr}-k*Q_{tr}}=dt$$ And by integrating both sides of the equation over time we are getting: $$\left(\frac{1}{k}\right) * ln \left(f_{tr}-k*Q_{tr}\right) \int_0^t= t \int_0^t $$ Which gives $$\left(\frac{1}{k}\right) * ln \left(\frac{f_{tr}-k*Q_{tr(t=t)}}{f_{tr}-k*Q_{tr(t=0)}}\right) = t $$ And again, Rearranging: $$Q_{tr(t=t)}=Q_{tr(t=0)}*e^{-kt}+\frac{f_{tr}}{k}*\left(1-e^{-kt}\right)$$
Bear in mind that the natural environment is usually a "pseudo steady state", the environmental properties change in time but are kept between thresholds. This means that the influx and outflux are equal and therefore can be calculated using the following formulas give us a good first order approximation of the fluxes. $$f_{tr}=R_{tr}$$ or $$\frac{d{tr}}{dt}=0$$ or $$\frac{dQ}{dT}=0$$
