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6.4 Formulation of the RPM-3D

The equations describing the transport of chemical species in RPM-IV are modified accordingly to accommodate the three dimensional description of the plume. Thus, Equation 6.4 is modified as follows:


 \begin{displaymath} \frac{d c_{j,k}^i}{d t} = \left(\frac{d c_{j,k}^i}{d t}\righ... ...c{d h_{j,k}}{d s}\right)uc^i_{j,k} + F^i_{Y,j,k} + F^i_{Z,j,k} \end{displaymath} (6.9)

where $c^i_{j,k}$ is the concentration of the $i$th species in the cell $(j,k)$; $j$ denotes the cell number in the horizontal direction, and $k$ denotes the cell number in the vertical. $F^i_{Y,j,k}$ and $F^i_{Z,j,k}$ denote the fluxes along the horizontal and vertical directions respectively. The equations for the calculation of the fluxes, are similar to Equations 6.5 and  6.6, and are as follows:

 
$\displaystyle F^i_{Y,j,k}$ $\textstyle =$ $\displaystyle E^i_{Y,j,k} - D^i_{Y,j,k}$  
$\displaystyle F^i_{Z,j,k}$ $\textstyle =$ $\displaystyle E^i_{Z,j,k} - D^i_{Z,j,k}$  
$\displaystyle E_{Y,j,k}^i$ $\textstyle =$ $\displaystyle \frac{1}{y_{j,k} - y_{j-1,k}} \left\{\left(\frac{d y_{j,k}}{d t}\right)c^i_{{j+1,k}} - \left(\frac{d y_{j-1,k}}{d t}\right)c^i_{j,k}\right\}$ (6.10)
$\displaystyle D_{Y,j,k}^i$ $\textstyle =$ $\displaystyle \frac{2}{y_{j,k} - y_{j-1,k}} \left\{K_{Y,j,k}\left(\frac{c^i_{j+... ...,j-1,k}\left(\frac{{c_{j,k}^i}-c^i_{j-1,k}}{y_{j,k} - y_{j-2,k}}\right)\right\}$  
$\displaystyle E_{Z,j,k}^i$ $\textstyle =$ $\displaystyle \frac{1}{z_{j,k} - z_{j,k-1}} \left\{\left(\frac{d z_{j,k}}{d t}\right)c^i_{{j,k+1}} - \left(\frac{d z_{j,k-1}}{d t}\right)c^i_{j,k}\right\}$  
$\displaystyle D_{Z,j,k}^i$ $\textstyle =$ $\displaystyle \frac{2}{z_{j,k} - z_{j,k-1}} \left\{K_{Z,j,k}\left(\frac{c^i_{j,... ...,j,k-1}\left(\frac{{c_{j,k}^i}-c^i_{j,k-1}}{z_{j,k} - z_{j,k-2}}\right)\right\}$  

where $y_{j,k}$, and $z_{j,k}$ denote the distances from the plume centerlines to the far most sides, in the horizontal and vertical directions respectively. $K_{Z,j,k}$ and $K_{Y,j,k}$ are solved using recursive relationships similar to Equation 6.7: the relationships are obtained from the following conditions: (a) $F^I_{Y,j,k}$and $F^I_{Z,j,k}$ are equal to zero for an inert species $I$, (b) there is a zero concentration gradient at the plume centerline in both horizontal and vertical directions.

The RPM-IV uses the Gear algorithm [76] to solve the stiff set of equations describing the chemical reactions. In this method, the step size for each successive calculation is adjusted according to the local truncation errors. The same approach is followed in the RPM-3D.

next up previous contents
Next: 6.5 Case Studies Up: 6. CHARACTERIZATION AND REDUCTION Previous: 6.3 Formulation of The
Sastry S. Isukapalli
1999-01-19