Trumpet concentrator

A geometry for reflective concentrators is the trumpet concentrator [128], Fig. 2.11. This is actually a hyperboloid of revolution. Besides the CPC it is denoted as one of the maximum collection efficiency nonimaging concentrators. It is characterized by multitudinous reflections near the exit aperture. 

Trumpet concentrators are ideal concentrators both in 2D and 3D case. They also offer the advantage of concentrating not only internal but also external rays (those latter are concentrated to an annular shape around the exit aperture). Trumpet concentrators are characterized by a small material expenditure. External rays are not usable for detector matrices. 

Option (Valid) presents a graph of relative standard deviation (c.o.v.) versus concentration, with the relative residuals superimposed. This gives a clear overview of the performance to be expected from a linear calibration Signal = A + B Concentration, both in terms of (relative) precision and of accuracy, because only a well-behaved analytical method will show most of the residuals to be inside a narrow trumpet -like curve this trumpet is wide at low concentrations and should narrow down to c.o.v. = 5% and rel. CL = 10%, or thereabouts, at medium to high concentrations. Residuals that are not randomly distributed about the horizontal axis point either to the presence of outliers, nonlinearity, or errors in the preparation of standards. 

It is a well-known fact that the precision in trace analysis decreases with diminishing concentration in a similar way as it does with decreasing sample weight (Sect. 2.1). The dependency of the repeatability and reproducibility standard deviation on the concentration of analytes has been investigated systematically at first by Horwitz et al. [1980] on the basis of thousands of pieces of interlaboratory data (mostly from food analysis). The result of the study has been represented in form of the well-known Horwitz trumpet which is represented in Fig. 7.3. 

Fig. 7.3. The so-called Horwitz trumpet Dependency of the relative standard deviation, rsd(x) on the concentration (x)...

When the oxide is formed by anodizing in acid solutions and the sample is then left to rest at the OCP, some dissolution can occur. This process has been studied by a numbers of authors,70-75 especially in relation to porous oxides [cf. Section 111(4)]. It was found that pore walls are attacked, so that they are widened and tapered to a trumpet-like shape.70 71 Finally, the pore skeleton collapses and dissolves, at the outer oxide region. The outer regions of the oxide body dissolve at higher rates than the inner ones.9,19 The same is true for dissolution of other anodic oxides of valve metals.76 This thickness dependence is interpreted in terms of a depth-dependent vacancy concentration in the oxide75 or by acid permeation through cell walls by intercrystalline diffusion, disaggregating the microcrystallites of y-alumina.4... 

Figure 71-1 Relationship of Laboratory CV (as %) with analyte concentration as powers of 1C)—exp. (For example, 6 on the abscissa represents a concentration of 10-6 or 1 ppm.) Note the shape of the curves has been referred to as Florwitz s trumpet.

A paper published by Hall and Selinger [3] points out that an empirical formula relating the concentration (c) to the coefficient of variation (CV) is also known as the precision (cr). They derive the origin of the trumpet curve using a binomial distribution explanation. Their final derived relationship becomes equation 72-2 ... 

Some media outlets trumpeted the 1996 book Our Stolen Future,5 a compilation of mostly unverified observations and speculations, and a paper published in Science that presented startling results purportedly showing that tiny concentrations of some chemicals behaved as endocrine disruptors. Congress rushed legislation that requires billions of dollars to be spent to test chemicals that were regarded as safe except for the alleged estrogenic effects.5... 

Coefficient of variation of interlaboratory results as a function of sample concentration (expressed as g analyte/g sample). The shaded region has been referred to as the Horwitz trumpet" because Of the way it flares open. [From W. Horwitz, Evaluation of Analytical Methods Used for Regulation of Foods and Drugs." Anal. Chem. 1982, 54.67A]... 

Figure 9.3. The FIRAS horn a compound parabolic concentrator with a trumpet bell flare to reduce sidelobes.

Figure 4.32. Graphical summary of validation indicators for file VALIDl.dat. In this depiction, the typical trumpet form (A, E) of the c.o.v./CL curves is seen the fact that there is a spread toward the right-hand edge (C) suggests that the measurement errors grow in a slightly over-proportional fashion with the concentration. The narrowing down of the "trumpet at the lowest concentration (D) is a sign that due to the proximity to the LOD, the measurement distribution is not truly Gaussian but has the low side clipped. The data points for the lowest concentration (10 ng/ml) are off scale (-r31.2 -r65.9), while the fourth-smallest con-...

Flow line concentrator (trumpet) Nonimaging collector in which the reflecting wall follows the lines of vector flux from a Lambertian source. 

FIGURE 18 (a) Practical thermal secondary concentrator referred to as the trumpet, which has been built and tested at the University of Chicago, (b) Profiles for a DCPC and two dielectric compound hyperbolic concentrators (DCHCs) used for photovoltaic secondaries. 

O Gallagher, J., and Winston, R. (1986). Test of a trumpet secondary concentrator with a paraboloidal dish primary, Solar Energy 36, 37. 

The target value for the standard deviation, o, should be circulated in advance to the PT participants along with a summary of the method by which it has been established. It will vary with anal)4e concentration, and one approach to estimating it is to use a functional relationship between concentration and standard deviation. The best-known relationship is the Horwitz trumpet, dating from 1982, so called because of its shape. Using many results from collaborative trials, Horwitz showed that the relative standard deviation of a method varied with the concentration, c (e.g. mg g" ), according to the approximate and empirical equation ... 

Note again the appearance of the 2 in equation (4.17), because two sample materials are studied. Here it is a simple matter to calculate that the estimate of si is (3.615). The mean of all the measurements is 49.33/2 = 24.665, so the relative standard deviation is (100 x 5.296)/24.665 = 21.47%. This seems to be a high value, but the Horwitz trumpet relationship would predict an even higher value of ca. 28% at this concentration level. It should be noted that possible outliers are not considered in the Youden procedure, so the possibility of rejecting the results from laboratory 1 does not arise. 

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