Question 4.13: Transient Species Concentrations in Open System with Linear ......
Transient Species Concentrations in Open System with Linear Chemical Reaction
Consider a sudden pollutant input into a body of water with inflow and outflow streams, or a continuous flow chemical reactor. Set up the compartmental model and solve the system for two species input modes: impulse and step functions at time t = 0.
\rm\frac{dc_i}{dt} =\sum\limits_{j=1}^{N}{(q_{ji}-q_{ij})+S_i(t)} \qquad\qquad\qquad(4.59)| Concept | Assumptions | Sketch |
| • Species mass m = ρ∀; ∀ = constant | • Well-mixed compartment with ∀ = constant |  |
| • Rate equation (4.59) with κ = 1 | • Reaction rate is constant | |
| • Constant flow rate |
Equation (4.59) can be rewritten in terms of species mass, m = ρ∀, as:
In light of the assumptions (see Sketch), Eq. (E.4.13.1) reduces to:
Where ∀/Q ≡ τ is the “residence time”, i.e., the theoretical duration that a fluid element (or particle) resides in the compartment.
Species Input ➀ (Impulse): At t = 0 a finite amount of, say, a pollutant is dumped into a lake (compartment), i.e., the lake water starts with an initial contaminant concentration \rm c_0 > 0. No species enters the lake thereafter, i.e., \rm c_{in} (t) = 0 because Δt_{input}≈ 0.
Species Input ➁ (Step Function): At t = 0 suddenly a fixed amount of a pollutant is discharged into the lake and that species input, \rm c_0 > 0 and \rm c_{in}(t) = c_{in} = c_0, stays constant.
The solutions to Eq. (E.4.13.2) for the two different input conditions are of the form of Eq. (4.62). Specifically, for Case ➀ we have with \rm c_0 > 0 and \rm c_{in}(t) = 0, while chemical reaction can be neglected, i.e., k = 0:
\rm c_i(t)=\sum\limits_{k=1}^{n}{A_ke^{-a_kt}} (4.62)
\rm c(t)=c_0~e^{-t/\tau}\qquad\qquad\qquad(E.4.13.3)And for Case ➁ with \rm c_{in} (t) = c_0 = constant (see Polyaminal and Zaitsev 1995):
Graph:
Comments:
• Clearly, with an initial amount of pollutant residing in the lake, i.e., \rm c(t = 0) = c_0 = 5 [M/ L^3 ] in Case 1, and no additional input, the contaminant is (exponentially) washed out via \rm Q_{out} .Note the pollutant residence time τ \sim Q_{out}^{-1} .
• In Case 2, due to the constant species input (i.e., \rm c_{in} (t) = c_0 ), an equilibrium concentration is reached due to the interplay of influx, efflux, and uptake.
