Introduction and definitions
This is to look for a unification of concepts developed in theory. Our reference work is \cite{hydrology} however, this matter has been developed since the sixties in Chemical engineering. Notably, here, we want to bridge \cite{Nauman_1969} to recent work which milestone can be considered \cite{Botter_2011}. Let's start with definitions:
- \(V\) is the control volume which we are analysing
- \(t\) is the clock time, or simply the time (as measured by a clock) which in \cite{Nauman_1969} is called \(\theta\)
- Particles are injected into the the control volume at time \(t_{in}\)
- and exit the control volume at time \(t_{ex}\), called the exit time
- Either \(t_{in}\) and \(t_{ex}\) can be random variables.
- Throughout the paper will be always assumed that \(t_{in\ } \leq t \leq t_{ex}\)
- \(T:=t_{ex}-t_{in}\) is called travel time of the particles
- If we take a snapshot of the control volume at time \(t\) particles in the volume will have a residence time \(T_r:=t-t_{in}\)
Therefore, after these definitions, we have two distinct distribution to care of: the distribution of travel times and the distribution of residence times. Notably, if we collect outgoing particles at the boundary of the control volume, i.e. the discharge (or evapotranspiration that we neglect here for simplicity), and we have their injection time distribution, for them \(t_{ex}=t\) and we are measuring their travel time.
The definition of residence times opens also to a new definition. Particles inside the control volume can have a
- life expectancy: \(T_e:=t_{ex}-t\), a random variable which will also have a distribution.
- Let \(D\) be the discharge.
Let's assume then that we can label each particle with its injection time and exit time (because eventually we found a way to forecast future). We can define then, relative to the control volume, the quantity:
- \(v(t,t_{in},t_{ex})\) being the volume of particles inside the control volume
We can normalize it over the total volume \(\)\(V\) to obtain:
- \(p(t_{in},t_{ex}|t):=v(t,t_{in},t_{ex})/V(t)\) that, assumed that injection time and exit time are random variables is their joint probability conditional to the clock time
Based on the above probability, we can define the marginals:
- the probability of life expectancy: