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# 03-31 - Announcements Prelim 1 Quiz 4 Wednesday April 7...

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Announcements Prelim 1 Quiz 4: Wednesday April 7: – Response equations for CSTR and PFR (no dispersion) to pulse and continuous inputs of conservative materials.

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SIMPLIFIED MODELS FOR RIVERS, STREAMS, & AFR'S Recall continuity equation (point form): i i J r -∇• + harpoonrightnosp i i C u C E harpoonrightnosp + - i J harpoonrightnosp C t = also Ex. Consider our earlier model for BOD L in a stream. The simplifying assumptions were: 1. 2. Plug flow (no mixing). E = 0 Complete lateral mixing (width and depth averaging). 1-D problem.
The continuity equation in this case reduces to: ° this is exactly the same as the equation we had for BOD the 1-D part no dispersion for first order kinetics ( 29 29 ( 29 29 x x i C J i r t X = - + arrowrightnosp arrowrightnosp ( 29 29 x i u C r X = - + = - C C u kC t X = - - so in the plug flow model. dL = -u dL -k r L dt dx

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We solved this at steady-state ( dL = ) dt 0 with boundary conditions of L = L O @ X = 0. The point here is that the continuity equation gives the same result, given the same simplifying assumptions. previously used so 0 O L x r L dL k dX L u = - r O L X ln k L u = - r X k u O L L e - = and We used the same assumptions for D.O. in the stream, however, r i also included a reaeration term: 1 2 dC dC u k L k (C* C) dt dX = - - + - + -
Predicted results (for BOD L ) @ steady-state with continuous discharge were: r r w w o r w Q L Q L L Q Q + = + + L O ) u / X ( r k o x e L L - = mass of the pulse input Let W' = the mass rate of pollutant discharge. w w r r L Q L Q ' W + = C and total W' Q = ______________________________________________________ For a pulse input (instantaneous discharge) of waste @ X=0 and t=0. If the pulse results in a concentration C O @ X=0 then this concentration will move as a plug downstream with velocity U and will decay with first order kinetics.

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