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29.6.1 Computing Volume Integrals



Volume


The volume of a surface is computed by summing the volumes of the cells that comprise the zone:


 \int { dV } = \sum_{i=1}^{n}{ \vert V_i \vert } (29.6-1)



Sum


The sum of a specified field variable in a cell zone is computed by summing the value of the selected variable at each cell in the selected zone:


 \sum_{i=1}^{n}{ \phi_i } (29.6-2)



Volume Integral


A volume integral is computed by summing the product of the cell volume and the selected field variable:


 \int { \phi dV } = \sum_{i=1}^{n}{ \phi_i \vert V_i \vert } (29.6-3)



Volume-Weighted Average


The volume-weighted average of a quantity is computed by dividing the summation of the product of the selected field variable and cell volume by the total volume of the cell zone:


 \frac{1}{V} \int{\phi dV} = \frac{1}{V} \sum_{i=1}^{n}{\phi_i \vert V_i \vert } (29.6-4)



Mass-Weighted Integral


The mass-weighted integral is computed by summing the product of density, cell volume, and the selected field variable:


 \int { \phi \rho dV } = \sum_{i=1}^{n}{ \phi_i \rho_i \vert V_i \vert } (29.6-5)



Mass-Weighted Average


The mass-weighted average of a quantity is computed by dividing the summation of the product of density, cell volume, and the selected field variable by the summation of the product of density and cell volume:


 \frac {\displaystyle{\int}{\phi \rho dV}}{\displaystyle{\int... ...vert}} {\displaystyle{\sum_{i=1}^{n}} {\rho_i \vert V_i\vert}} (29.6-6)


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