Explicit Matrix Solution for Total Exchange Areas

Explicit Matrix Solution for Total Exchange Areas For gray or monochromatic transfer, the primary working relation for zoning calculations via the matrix method is 

Equation (5-118) makes full allowance for multiple reflections in an enclosure of any degree of complexity. To apply Eq. (5-118) for design or simulation purposes, the gas temperatures must be known and surface boundary conditions must be specified for each and every surface zone in the form of either E, or Q,. In application of Eq. (5-118), physically impossible values of E, may well result if physically unrealistic values of Q, are specified.  

(5-118), SS and SG are defined as the required arrays of total surface-to-surface exchange areas and total gas-to-surface exchange areas, respectively. The matrices for total exchange areas are calculated explicitly from the corresponding arrays of direct exchange areas and the other enclosure parameters by the following matrix formulas  

While the R matrix is generally not symmetric, the matrix product pI R is always symmetric. This fact proves useful for error checking. 

The most computationally significant aspect of the matrix method is that the inverse reflectivity matrix R always exists for any physically meaningful enclosure problem. More precisely R always exists provided that K 0. For a transparent medium, R exists provided that there formally exists at least one surface zone A, such that c, 0. An important computational corollary of this statement for transparent media is that the matrix [Al — ss] is always singular and demonstrates 

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Explicit Matrix Solution for Total Exchange Areas. 5-25... 

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