ANSYS FLUENT 12.0 Theory Guide - 20.4.1 Computing Volume Integrals

20.4.1 Computing Volume Integrals

Volume

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

$\int { dV } = \sum_{i=1}^{n}{ \vert V_i \vert }$

(20.4-1)

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:

$\sum_{i=1}^{n}{ \phi_i }$

(20.4-2)

Volume Integral

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

$\int { \phi dV } = \sum_{i=1}^{n}{ \phi_i \vert V_i \vert }$

(20.4-3)

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:

$\frac{1}{V} \int{\phi dV} = \frac{1}{V} \sum_{i=1}^{n}{\phi_i \vert V_i \vert }$

(20.4-4)

Mass-Weighted Integral

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

$\int { \phi \rho dV } = \sum_{i=1}^{n}{ \phi_i \rho_i \vert V_i \vert }$

(20.4-5)

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:

$\frac {\displaystyle{\int}{\phi \rho dV}}{\displaystyle{\int... ...vert}} {\displaystyle{\sum_{i=1}^{n}} {\rho_i \vert V_i\vert}}$

(20.4-6)

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