Transport and Reaction Equation IPOLA Equation The fundamental mass balance equation is A L O P I ( 0 ) where: I = inputs P = production O = outputs L = losses A = accumulation Consider a 3-D ‘cell’ like this where J x | x indicates the flux density in the x direction at the point x. Fluxes into the volume represent Inputs (I), while fluxes out of the volume are Outputs (O). Production (P), Losses (L), and Accumulation (A) inside the volume are also possible. The cell can represent a small piece of many different things; it might represent a volume of water in a pond, part of the atmosphere, a portion of an aquifer, or a block of steel. x z J x | x J x | x+ x J z | z J z | z+ z y J y | y J y | y+ y P L A
Similarly, the flux density J might be a liquid flow, a heat (energy) flux, or a chemical flux. Since our primary interest is in the movement of chemicals in the environment, we’ll consider only the flux of chemical mass here. The approach has broad applicability though. Advective Flux Density The mass flux (ML -2 T -1 ) can consist of a number of components. First, there may be a fluid (liquid or gas) flow with velocity v that carries along a mass of chemical. Let’s consider just the x direction. In time t, a velocity v (at x) will sweep in a volume V = v| x t y z. The volume swept out may be different because it depends on the velocity at x + x. The chemical mass going in or out in time t depends on the fluid volume and the concentration C in the fluid: Mass/Time = Concentration x Volume/Time. If we divide everything by the area of the inflow (or outflow) faces, we get the mass flux density, which ends up being very simple: J x | x = C v where C is concentration and v is the velocity at point x.
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