Why is this important?
There are many applications in which it is desirable to map the concentration of a gas species, e.g. the fuel-air distribution within a combustion chamber, the distribution of soot within a flame, or CO2 within the exhaust of a gas turbine. Measuring gas concentrations using probes is often impractical because the probe may perturb the concentration field, or the field may be physically-inaccessible. Moreover, physical probes offer very limited temporal and spatial resolution, which makes them unsuitable for turbulent flow fields. Chemical species tomography overcomes these drawbacks.
How does CST work?
In CST the concentration of a target species is tomographically-reconstructed from radiation intensity measurements made along the periphery of the flow field. This measurement exploits the fact that gas molecules absorb radiation at specific wavelengths, λ, or wavenumbers, η = 1/λ.
A laser is shone across the flow field. The ratio of incident and transmitted intensity is related to the distribution of the spectral absorption coefficient, κη along the corresponding line-of-sight (LOS) by the Beer-Lambert law
where u is a parametric distance along the beam. In the case of an isothermal and isobaric flow, the spectral absorption coefficient is proportional to the target species concentration. Inferring κη(u) from a single LOS measurement is mathematically ill-posed.
Most often the domain is discretized into n pixels, in which case the intensity is modeled as
where xi is the spectral absorption coefficient in the ith pixel, bi = ln(Iη,0/Iη,L,i) and Aij is the chord length of the ith beam subtended by the jth pixel.
Writing this equation for a set of m laser beams results in an (m X n) matrix equation, Ax = b, but the number of pixels far exceeds the number of beams. Consequently, this matrix is rank-deficient, and an infinite number of solutions {x} exist that could explain the measurement data. The smallest of these solutions, xLS, is nonphysical.
Example beam arrangement
The smallest norm solution to Ax = b is nonphysical
Prior information
In order to solve this inverse problem, the measurement equations must be supplemented with prior information about the flow. This information could specify a certain degree of spatial smoothness, or a mean and spatial covariance derived from turbulent flow physics. Prior information and measurement data is combined using Bayesian inference.
Turbulent CH4 jet
Ground truth
Smoothness prior
Turbulence theory
Optimal beam arrangement
How should the beams be arranged to maximize the information content of the data? This question can be answered using Bayesian design-of-experiment theory, in which the optimal beam arrangement minimizes an objective function that scales with the posterior covariance. This procedure accounts for the prior knowledge about the flowfield.
Strong correlation between objective function and reconstruction error
Optimized beam arrangement
Ground-truth phantom
Reconstruction with optimal beam array
Other tomography modalities
CST can be applied to other tomography modalities, including broadband absorption tomography, emission tomography, in which intensity measurements are used to reconstruct a light source (incandescence, chemiluminesence), hyperspectral absorption tomography (combined absorption and temperature, multiple species), and background-oriented Schlieren (BOS) tomography, which reconstructs the 3D time-resolved refractive index of the field. The refractive index is correlated to temperature.
BOS tomography setup, Prof Khadijah Mohri, IVG, University of Duisburg-Essen
Instantaneous 3D reconstruction of a methane flame
