R

z), K(z,r) = 2r/(r2-z2)1/2 the Parameter relating optical pathlength to radial location, r is the radial distance from the center-point, and f (r) = -a- Pabs - XCH 4.

The solution to this equation is not trivial. Typically, this equation is solved by reconstruction techniques such as an Abel inversion or Fourier deconvolution [17,18]. Looking at Eqn. 3 it is seen that if we knew the concentration profile it would be relatively easy to calculate the LOS intensity ratios that would be generated from it. This is accomplished by breaking the LOS measurement into segments at a small enough scale that assuming the concentration across a segment is constant would result in little error. Eqn. 6 shows an example for a measurement line:

where Dl i is the distance across segment i and XCH4 i is the averaged concentration over segment i.

For this research we have come up with a GA to reconstruct the time averaged concentration profile of an axis-symmetric jet from LOS measurements. Additionally, we modified the program to allow reconstruction of the RMS concentration profile from RMS values of the intensity ratios, and further modified the code for an asymmetric jet. Kihm et al. [19] have applied a genetic algorithm based program to the case of tomographic reconstruction of a time-averaged concentration field and found that it performed well with a small number of measurements. In the interest of brevity, we refer the reader to Powel and Skolnick [20] and Kihm et al. [19] for a discussion of the principles of GAs.

The reconstruction of radial concentration profiles (mean and RMS) from our experimental line of sight measurements is desired. In order to test the ability of the GA to reconstruct the time-averaged and RMS concentration profiles, two sets of data were used where the correct answer was known a priori. The first set of test data assumed that the fuel concentration varied as a function of radial position with the form of an inverted parabola. The equation for this concentration profile was known, and thus the solution to Eqn. 1 can be found analytically. For these cases, the intensity ratios for the absorption lines can be found for the concentration profile and the results of the GA can then be compared to the known answer. The results of this comparison are shown in Fig. 2. As can be seen from the figure, the genetic algorithm performed satisfactorily at reconstructing the concentration field for a parabolic input.

The GA program was modified to reconstruct the RMS of the radial concentration profile based on the RMS of the measured LOS intensity ratios. The theory involved in this procedure is that of propagation of error. Eqn. 6 describes the relation between the natural log of the intensity ratio for a given absorption line and the concentration field. Thus the natural log of the intensity ratio, ln(I/Io), is a linear function of the time-dependent concentrations of each radial position. The RMS of ln(I/Io) then follows Equation 7 (adapted from Beckwith et al. [21]).

Equation 7 assumes that the time-varying concentration at each radial position is independent of the time-varying concentration at every other radial position. If this independence does not exist, there will be correlation terms, and the relation between RMS of ln(I/Io) will no longer be as in eqn. 7. We can apply the same type of program to Eqn. 7 to reconstruct the RMS concentration profile given the RMS values of ln(I/Io). Because our sampling rate for the LOS measurements was high enough to see the time-dependent features of the flow (sampling rate was 10 kHz), we can determine the RMS values of ln(I/Io) based on fluctuations whose periods are 200 m seconds or greater.

We modified the parabolic concentration profile to include a sinusoidal fluctuation in order to test the capability of the genetic algorithm scheme for reconstruction of the RMS of the concentration field. The equation for the time-dependent concentration field is then of the form:

where X(r,t) is the time-dependent concentration, VA the Amplitude (previous, non-time-dependent concentration field) and w the frequency. The square of the sinusoidal function was used so the concentration would not take on negative values. The RMS of X(r,t) is then found analytically to be

We therefore know the RMS of the concentration field analytically, and it is also easy to find the RMS of the intensity ratio for the given absorption lines for this case, because if X(r,t) follows the form of Eqn. 8, one can integrate Eqn. 3. Thus, we can check the performance of the GA for reconstructing the RMS of the concentration profile for this case. These results are also shown in Fig. 2.

It can be seen from Fig. 2 that the GA successfully reproduces the right overall trend for the profile of the concentration RMS. However, small departures from the actual RMS values are noticeable. This departure is due to the fact that the time-dependent concentration value at a given radial position is correlated to all other positions, so that there will be some error in applying Eqn. 7. This was done intentionally as a test of how well this reconstruction scheme would handle correlated data. Our analytical test showed that the correlated data gave roughly the same answer as uncorrelated data to within 30%. With the current treatment of the problem of reconstructing the signal RMS, we find no direct way of overcoming error due to the interdependence of the concentration values. A method that would remove this problem would be to use the time dependent data, reconstructing the concentration field at each time step from the experimental results at that time step, and repeating this for around 200 time steps.

A second set of numerical tests was done to further explore the capabilities of the GA program to reconstruct the concentration field. These tests consisted of generating a set of time-resolved LES results, and calculating the RMS and time-averaged fuel concentration profile. From these, the intensity ratios that would give the concentration profile were found at each time step using Eqn. 6, then the RMS of the intensity ratio was calculated from these values. 200 time steps of LES output data were used. The intensity ratios for the time-averaged concentration profile were used as an input to the genetic algorithm, and the results are given in Fig. 3. From Fig. 3 we see that the GA once again performs satisfactorily at reconstructing the radial profile of time-averaged concentration, with the center-point concentration slightly underpredicted.

Fig. 3 also gives the reconstructed profile of the concentration RMS. Once again, the general trends are represented and the values are correct to within 30% of the actual values. Again the departures are felt to be due to the fact that the time-varying concentration values at each radial position are not completely independent. The results of the GA tend to give adequate agreement for the time-averaged profiles and modest agreement for the RMS profiles (Figs. 2 and 3). Therefore the GA technique was employed for quantitative evaluation of the LES model for time-averaged data, and for semi-quantitative evaluation of the LES for the time-dependent results.

We then took the additional step of modifying the program to enable reconstruction of an asymmetric concentration field. This would of course require more measurements - taken at more than one angular position at the pipe exit. A Gaussian function with a peak at 0.13 was used for the concentration field, but with the peak of the Gaussian shifted off-center by 0.5 cm. The 2-D version of the code uses multiple angles of data (6 angles in the present state) corresponding to 60 input measurements to optimize. The results of using line of sight data that would correspond to a shifted Gaussian curve (not shown) demonstrated that given enough data points, this method can successfully be used to reconstruct an asymmetric concentration profile. Future work may apply this program to 2-dimensional sets of measurement data.

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