Annex 5: Residence times in coarse gravel filtration units A5.1. Modelling residence-time distribution. A useful model for residence time distribution is shown in figure A5-1, in which virtual completely mixed (CM) reactors in series are used to simulate hydraulic behaviour of real reactors. The number of CM reactors can have any integer value from 1,0 to n. In general, the sum of the volumes of all the virtual reactors is equal to the volume of the real system being modelled. V = V 1 + V 2 + V 3 + ................. + V n (A5-1) Figure A5-1. Virtual CM reactors in series for modelling the hydraulic performance of real reactors. As shown in the figure A5-1, at time t o , a slug or step dose of tracer is added to the influent side of the CM reactors. The tracer concentration C n at the effluent side of the CM reactors is given by the equation A5-2 (Nauman and Buffham, 1983, quoted by Clark, 1996; Galvis and Perez, 1985), C n/C o = F(t) = n n−1 [(nθ) /(n − 1)! ] e −nθ (A5-2) In which C n , is referred to the maximum expected concentration Co in the last reactor; F (t) is the cumulative residence time distribution; θ = t/To, being To equal to the total volume of reactor (V) divided by the flow (Q); and n, the number of CM reactors in series. Equation A5-2 is a one-parameter (for n) model. As shown in figure A5-2, equation A5-2 allows obtaining curves of dimensionless residence time density versus θ, dimensionless time. As the number reactors increases, the residence time density moves from the exponential distribution of the single perfectly mixed tank (n = 1) to a distribution that increasingly seems to be centred at θ = 1. Therefore, as n approaches ∞, the residence-time density for the reactors in series approaches the residence-time density for ideal plug flow. This is one of the chief strengths of the reactors in series model. Using equation A5-2, a general expression (equation A5-3) can be obtained for calculating 1 – F (t), which is the fraction of the flow that remains in the reactor system for a period longer than t: 1- F(t) = n (nθ) ∑ e (i − 1)! i=1 i-1 .-nθ (A5-3) Applying equation A5-3 for different number (n) of reactors, it can be obtained figure A5-3 and table A5-1, that show the relation between 1-F (t) and θ and the percentages of flow discharges at different fractions θ respectively. A5-1
Figure A5-2 Residence time density for different number (n) of CM reactors in series Figure A5-3 Residence time characteristics of CM reactors in series. A5-2
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