Uncertainties propagation for combustion diagnostics using infrared band models

The development of matrix equations for an uncertainty propagations analysis of an emission-absorption (E/A) band-model technique is described. The E/A technique is used for the determination of local values of temperature and species concentration from measurements of radiance and transmittance on axisymmetric gas flows. The approach used depends on linearization of the original nonlinear system of equations comprising the solution to the radiative transfer problem for an emitting-absorbing source such as a turbojet or rocket engine exhaust plume. The resultant linear transformation matrix provides a means for determination of the variance-covariance matrix for inverted temperature and partial pressure, based on the experimentally determined variance-covariance matrix of the measured radiance and transmittances. The diagonal elements of the propagated variance-covariance matrix yield the square of the standard deviation of the temperature and partial pressure. Although the equations are developed for uncertainty propagations analysis of the E/A band-model technique, the results are a general property of linear transformations. They are thus applicable to any situation in which the dependence of analysis on measurements can be approximated by a linear relationship. Inversion of radiance and transmittance lateral profiles obtained from a liquid-propellant rocket engine to radial profiles of temperature and partial pressure with the associated uncertainties propagations is included as an illustrative example.