Mass Balance in an Infinitely Small Control Volume The Advection-Dispersion-Reaction Equation

5 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 

The only difference between Eqs. [1-5] and [1-la] and [1-lb] is that, because the control volume is of an unspecified, arbitrarily small size, each term is expressed as mass per unit time per unit volume. Thus, the leftmost term, dC/dt, represents the rate at which a chemical s concentration (storage per unit time) changes at a fixed point in a flowing fluid. The concentration 

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  

Note that the transport terms (the second and third terms) in Eq. [1-6] are the three-dimensional counterparts of the corresponding terms in Eq. [1-5], As in Eq. [1-4], D is assumed equal in all directions. In many cases, this assumption is an oversimplification the value of D in the direction of flow can be very different than the value perpendicular to flow (i.e., D may be anisotropic). Furthermore, D may vary with location (i.e., be inhomogeneous), or vary with time. Often, a larger value of D may become applicable as the scale of the problem increases. 

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