Waterloo Laboratory for Inverse Analysis and Thermal Sciences (WATLIT)

295 Phillip Street

Waterloo, Ontario, Canada

N2L 3W8

### 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 CO_{2} 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 κ_{η}(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 x_{i} is the spectral absorption coefficient in the ith pixel, b_{i} = ln(I_{η,0}/I_{η,L,i}) and A_{ij} 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, **x**_{LS}, 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.

### 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.

### 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.