Advection-dispersion-reaction

MASS BALANCE IN AN INFINITELY SMALL CONTROL VOLUME THE ADVECTION-DISPERSION-REACTION EQUATION... 

So far, the concept of mass conservation has been applied to large, easily measurable control volumes such as lakes. Mass conservation also can be usefully expressed in an infinitesimal control volume, mathematically considered to be a point. Conservation of mass is expressed in such a volume with the advection—dispersion-reaction equation. This equation states that the rate of change of chemical storage at any point in space, dC/dt, equals the sum of both the rates of chemical input and output by physical means and the rate of net internal production (sources minus sinks). The inputs and outputs that occur by physical means (advection and Fickian transport) are expressed in terms of the fluid velocity (V), the diffusion/dispersion coefficient (D), and the chemical concentration gradient in the fluid (dC/dx). The input or output associated with internal sources or sinks of the chemical is represented by r. In one dimension, the equation for a fixed point is... 

Equation [1-5] is pertinent to a one-dimensional system, such as a long, narrow tube full of water, where significant variations in concentration may be assumed to occur only along the length of the tube. In a three-dimensional situation, the advection-dispersion-reaction equation can be represented most succinctly using vector notation, where V is the divergence operator ... 

Mass balance within an arbitrarily chosen biofilm section, or slice, taken parallel to the surface of attachment, is described by the one-dimensional, advection-dispersion-reaction equation, Eq. [1-5], with steady-state conditions and no advection. The sink term is microbial uptake, modeled using the parameters discussed in Section 2.6.3 see Eqs. [2-71 a] and [2-72],... 

The smallest spatial scale at which outdoor air pollution is of concern corresponds to the air volume affected by pollutant chemical emissions from a single point source, such as a smokestack (Fig. 4-24). Chemicals are carried downwind by advection, while turbulent transport (typically modeled as Fick-ian transport) causes the chemical concentrations to become more diluted. Typically, smokestacks produce continuous pollutant emissions, instead of single pulses of pollutants thus, steady-state analysis is often appropriate. At some distance downwind, the plume of chemical pollutants disperses sufficiently to reach the ground the point at which this occurs, and the concentrations of the chemicals at this point and elsewhere, can be estimated from solutions to the advection-dispersion-reaction equation (Section 1.5), given a knowledge of the air (wind) velocity and the magnitude of Fickian transport. 

If the Fickian transport coefficient is known, it is possible to predict the distribution of the tracer at any time and location after it is introduced into the column. At the time of injection of the tracer (f = 0), the concentration is high over a short length of column. At a later time fi, the center of the mass of tracer has moved a distance equivalent to the seepage velocity multiplied by fi, and the mass has a broader Gaussian, or normal, distribution, as defined in Eq. (2.6). For this one-dimensional situation, the solution to the advection-dispersion-reaction equation (Eq. 1.5) gives the concentration of the tracer as a function of time and distance. 

FIGURE 3.19 Solutions to the advection-dispersion-reaction equation (Eq. 1.5) for an ideal tracer. Cases for continuous input of mass beginning at time f = 0 are adapted from references cited, assuming x and/or r are much larger than D/v r equals x +y in two dimensions or... 

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