Advection-Diffusion Equation

1D Water Quality Solver models the transport of pollutants along open channel reaches by the one-dimensional advection-diffusion equation,

                                       (1)

Where:
C = pollutant concentration (kg/m3)
A = cross-sectional flow area (m2)
u = cross-sectionally averaged flow velocity (m/s)
D = diffusion coefficient (m2/s)
X = distance (m)
t = time (s)
S = source/sink term; representing decay, growth, erosion, deposition, etc (kg/m/s).

As the equation is one dimensional all the variables represent cross-sectionally averaged quantities.

Equation 1 is effectively a mass conservation equation with an added source term, S. The first term represents the rate of change of pollutant at a point. The second term is called the advection component and when combined with the first term represents the rate of change of pollutant in a unit of fluid along a streamline (considering the carrying fluid as incompressible). This is then balanced by the third term, the diffusion term, which represents the flux of pollutant out of a small unit of fluid travelling with the flow.

Due to the complexity of the river network boundary conditions, Equation 1 is solved numerically. The finite difference approximation to Equation 1 used in the 1D Water Quality Solver is:

                                       (2)

Where:
N = time index
I = position index
Δx = the mean of the element lengths adjacent to node I (m)
Δt = time step (s)
Φ = C x A = scalar transport variable (kg/m)

Equation 2 introduces a numerical problem as we now need to determine the likely concentrations at sub-nodal values as required by the second term in Equation 2.

A number of standard methods are available for doing this and will be discussed in the next section. 1D Water Quality Solver gives the user a choice of three algorithms; Upwinding, QUICK and SMART. It is recommended that SMART is used. The SMART algorithm is the algorithm, which is used by 1D Water Quality Solver by default. In order for the user to switch to using the Upwinding and the QUICK algorithms the user has to contact Flood Modeller Customer Support for the details on how to do this.