Computing Fresnel integrals via modified trapezium rules

In this paper we propose methods for computing Fresnel integrals based on truncated trapezium rule approximations to integrals on the real line, these trapezium rules modified to take into account poles of the integrand near the real axis. Our starting point is a method for computation of the error function of complex argument due to Matta and Reichel (J Math Phys 34:298–307, 1956) and Hunter and Regan (Math Comp 26:539–541, 1972). We construct approximations which we prove are exponentially convergent as a function of N , the number of quadrature points, obtaining explicit error bounds which show that accuracies of 10−15 uniformly on the real line are achieved with N=12 , this confirmed by computations. The approximations we obtain are attractive, additionally, in that they maintain small relative errors for small and large argument, are analytic on the real axis (echoing the analyticity of the Fresnel integrals), and are straightforward to implement.

Steady-state thermal analysis of an innovative receiver for linear Fresnel reflectors

The study of the performance of an innovative receiver for linear Fresnel reflectors is carried out in this paper, and the results are analyzed with a physics perspective of the process. The receiver consists of a bundle of tubes parallel to the mirror arrays, resulting on a smaller cross section for the same receiver width as the number of tubes increases, due to the diminution of their diameter. This implies higher heat carrier fluid speeds, and thus, a more effective heat transfer process, although it conveys higher pumping power as well. Mass flow is optimized for different tubes diameters, different impinging radiation intensities and different fluid inlet temperatures. It is found that the best receiver design, namely the tubes diameter that maximizes the exergetic efficiency for given working conditions, is similar for the cases studied. There is a range of tubes diameters that imply similar efficiencies, which can drive to capital cost reduction thanks to the flexibility of design. In addition, the length of the receiver is also optimized, and it is observed that the optimal length is similar for the working conditions considered. As a result of this study, it is found that this innovative receiver provides an optimum design for the whole day, even though impinging radiation intensity varies notably. Thermal features of this type of receiver could be the base of a new generation of concentrated solar power plants with a great potential for cost reduction, because of the simplicity of the system and the lower weigh of the components, plus the flexibility of using the receiver tubes for different streams of the heat carrier fluid.

SolarPaces 2013: Innovations on direct steam generation in linear fresnel collectors

Direct Steam Generation (DSG) in Linear Fresnel (LF) solar collectors is being consolidated as a feasible technology for Concentrating Solar Power (CSP) plants. The competitiveness of this technology relies on the following main features: water as heat transfer fluid (HTF) in Solar Field (SF), obtaining high superheated steam temperatures and pressures at turbine inlet (500ºC and 90 bar), no heat tracing required to avoid HTF freezing, no HTF degradation, no environmental impacts, any heat exchanger between SF and Balance Of Plant (BOP), and low cost installation and maintenance. Regarding to LF solar collectors, were recently developed as an alternative to Parabolic Trough Collector (PTC) technology. The main advantages of LF are: the reduced collector manufacturing cost and maintenance, linear mirrors shapes versus parabolic mirror, fixed receiver pipes (no ball joints reducing leaking for high pressures), lower susceptibility to wind damages, and light supporting structures allowing reduced driving devices. Companies as Novatec, Areva, Solar Euromed, etc., are investing in LF DSG technology and constructing different pilot plants to demonstrate the benefits and feasibility of this solution for defined locations and conditions (Puerto Errado 1 and 2 in Murcia Spain, Lidellin Newcastle Australia, Kogran Creek in South West Queensland Australia, Kimberlina in Bakersfield California USA, Llo Solar in Pyrénées France,Dhursar in India,etc). There are several critical decisions that must be taken in order to obtain a compromise and optimization between plant performance, cost, and durability. Some of these decisions go through the SF design: proper thermodynamic operational parameters, receiver material selection for high pressures, phase separators and recirculation pumps number and location, pipes distribution to reduce the amount of tubes (reducing possible leaks points and transient time, etc.), etc. Attending to these aspects, the correct design parameters selection and its correct assessment are the main target for designing DSG LF power plants. For this purpose in the recent few years some commercial software tools were developed to simulatesolar thermal power plants, the most focused on LF DSG design are Thermoflex and System Advisor Model (SAM). Once the simulation tool is selected,it is made the study of the proposed SFconfiguration that constitutes the main innovation of this work, and also a comparison with one of the most typical state-of-the-art configuration. The transient analysis must be simulated with high detail level, mainly in the BOP during start up, shut down, stand by, and partial loads are crucial, to obtain the annual plant performance. An innovative SF configurationwas proposed and analyzed to improve plant performance. Finally it was demonstrated thermal inertia and BOP regulation mode are critical points in low sun irradiation day plant behavior, impacting in annual performance depending on power plant location.

Towards cost reduction in concentrating solar power: innovative design for an efficient fresnel based solar field

Energía termosolar (de concentración) es uno de los nombres que hacen referencia en español al término inglés “concentrating solar power”. Se trata de una tecnología basada en la captura de la potencia térmica de la radiación solar, de forma que permita alcanzar temperaturas capaces de alimentar un ciclo termodinámico convencional (o avanzado); el futuro de esta tecnología depende principalmente de su capacidad para concentrar la radiación solar de manera eficiente y económica. La presente tesis está orientada hacia la resolución de ciertos problemas importantes relacionados con este objetivo. La mencionada necesidad de reducir costes en la concentración de radiación solar directa, asegurando el objetivo termodinámico de calentar un fluido hasta una determinada temperatura, es de vital importancia. Los colectores lineales Fresnel han sido identificados en la literatura científica como una tecnología con gran potencial para alcanzar esta reducción de costes. Dicha tecnología ha sido seleccionada por numerosas razones, entre las que destacan su gran libertad de diseño y su actual estado inmaduro. Con el objetivo de responder a este desafío se desarrollado un detallado estudio de las propiedades ópticas de los colectores lineales Fresnel, para lo cual se han utilizado métodos analíticos y numéricos de manera combinada. En primer lugar, se han usado unos modelos para la predicción de la localización y la irradiación normal directa del sol junto a unas relaciones analíticas desarrolladas para estudiar el efecto de múltiples variables de diseño en la energía incidente sobre los espejos. Del mismo modo, se han obtenido analíticamente los errores debidos al llamado “off-axis aberration”, a la apertura de los rayos reflejados en los espejos y a las sombras y bloqueos entre espejos. Esto ha permitido la comparación de diferentes formas de espejo –planos, circulares o parabólicos–, así como el diseño preliminar de la localización y anchura de los espejos y receptor sin necesidad de costosos métodos numéricos. En segundo lugar, se ha desarrollado un modelo de trazado de rayos de Monte Carlo con el objetivo de comprobar la validez del estudio analítico, pero sobre todo porque este no es preciso en el estudio de la reflexión en espejos. El código desarrollado está específicamente ideado para colectores lineales Fresnel, lo que ha permitido la reducción del tiempo de cálculo en varios órdenes de magnitud en comparación con un programa comercial más general. Esto justifica el desarrollo de un nuevo código en lugar de la compra de una licencia de otro programa. El modelo ha sido usado primeramente para comparar la intensidad de flujo térmico y rendimiento de colectores Fresnel, con y sin reflector secundario, con los colectores cilíndrico parabólicos. Finalmente, la conjunción de los resultados obtenidos en el estudio analítico con el programa numérico ha sido usada para optimizar el campo solar para diferentes orientaciones –Norte-Sur y Este-Oeste–, diferentes localizaciones –Almería y Aswan–, diferentes inclinaciones hacia el Trópico –desde 0 deg hasta 32 deg– y diferentes mínimos de intensidad del flujo en el centro del receptor –10 kW/m2 y 25 kW/m2–. La presente tesis ha conducido a importantes descubrimientos que deben ser considerados a la hora de diseñar un campo solar Fresnel. En primer lugar, los espejos utilizados no deben ser plano, sino cilíndricos o parabólicos, ya que los espejos curvos implican mayores concentraciones y rendimiento. Por otro lado, se ha llegado a la conclusión de que la orientación Este-Oeste es más propicia para localizaciones con altas latitudes, como Almería, mientras que en zonas más cercanas a los trópicos como Aswan los campos Norte-Sur conducen a mayores rendimientos. Es de destacar que la orientación Este-Oeste requiere aproximadamente la mitad de espejos que los campos Norte-Sur, puediendo estar inclinados hacia los Trópicos para mejorar el rendimiento, y que alcanzan parecidos valores de intensidad térmica en el receptor todos los días a mediodía. Sin embargo, los campos con orientación Norte-Sur permiten un flujo más constante a lo largo de un día. Por último, ha sido demostrado que el uso de diseños pre-optimizados analíticamente, con anchura de espejos y espaciado entre espejos variables a lo ancho del campo, pueden implicar aumentos de la energía generada por metro cuadrado de espejos de hasta el 6%. El rendimiento óptico anual de los colectores cilíndrico parabólicos es 23 % mayor que el rendimiento de los campos Fresnel en Almería, mientras que la diferencia es de solo 9 % en Aswan. Ello implica que, para alcanzar el mismo precio de electricidad que la tecnología de referencia, la reducción de costes de instalación por metro cuadrado de espejo debe estar entre el 10 % y el 25 %, y que los colectores lineales Fresnel tienen más posibilidades de ser desarrollados en zonas de bajas latitudes. Como consecuencia de los estudios desarrollados en esta tesis se ha patentado un sistema de almacenamiento que tiene en cuenta la variación del flujo térmico en el receptor a lo largo del día, especialmente para campos con orientación Este-Oeste. Este invento permitiría el aprovechamiento de la energía incidente durante más parte del año, aumentando de manera apreciable los rendimientos óptico y térmico. Abstract Concentrating solar power is the common name of a technology based on capturing the thermal power of solar radiation, in a suitable way to reach temperatures able to activate a conventional (or advanced) thermodynamic cycle to generate electricity; this quest mainly depends on our ability to concentrate solar radiation in a cheap and efficient way. The present thesis is focused to highlight and help solving some of the important issues related to this problem. The need of reducing costs in concentrating the direct solar radiation, but without jeopardizing the thermodynamic objective of heating a fluid up to the required temperature, is of prime importance. Linear Fresnel collectors have been identified in the scientific literature as a technology with high potential to reach this cost reduction. This technology has been selected because of a number of reasons, particularly the degrees of freedom of this type of concentrating configuration and its current immature state. In order to respond to this challenge, a very detailed exercise has been carried out on the optical properties of linear Fresnel collectors. This has been done combining analytic and numerical methods. First, the effect of the design variables on the ratio of energy impinging onto the reflecting surface has been studied using analytically developed equations, together with models that predict the location and direct normal irradiance of the sun at any moment. Similarly, errors due to off-axis aberration, to the aperture of the reflected energy beam and to shading and blocking effects have been obtained analytically. This has allowed the comparison of different shapes of mirrors –flat, cylindrical or parabolic–, as well as a preliminary optimization of the location and width of mirrors and receiver with no need of time-consuming numerical models. Second, in order to prove the validity of the analytic results, but also due to the fact that the study of the reflection process is not precise enough when using analytic equations, a Monte Carlo Ray Trace model has been developed. The developed code is designed specifically for linear Fresnel collectors, which has reduced the computing time by several orders of magnitude compared to a wider commercial software. This justifies the development of the new code. The model has been first used to compare radiation flux intensities and efficiencies of linear Fresnel collectors, both multitube receiver and secondary reflector receiver technologies, with parabolic trough collectors. Finally, the results obtained in the analytic study together with the numeric model have used in order to optimize the solar field for different orientations –North-South and East-West–, different locations –Almería and Aswan–, different tilts of the field towards the Tropic –from 0 deg to 32 deg– and different flux intensity minimum requirements –10 kW/m2 and 25 kW/m2. This thesis work has led to several important findings that should be considered in the design of Fresnel solar fields. First, flat mirrors should not be used in any case, as cylindrical and parabolic mirrors lead to higher flux intensities and efficiencies. Second, it has been concluded that, in locations relatively far from the Tropics such as Almería, East-West embodiments are more efficient, while in Aswan North- South orientation leads to a higher annual efficiency. It must be noted that East-West oriented solar fields require approximately half the number of mirrors than NS oriented fields, can be tilted towards the Equator in order to increase the efficiency and attain similar values of flux intensity at the receiver every day at midday. On the other hand, in NS embodiments the flux intensity is more even during each single day. Finally, it has been proved that the use of analytic designs with variable shift between mirrors and variable width of mirrors across the field can lead to improvements in the electricity generated per reflecting surface square meter up to 6%. The annual optical efficiency of parabolic troughs has been found to be 23% higher than the efficiency of Fresnel fields in Almería, but it is only around 9% higher in Aswan. This implies that, in order to attain the same levelized cost of electricity than parabolic troughs, the required reduction of installation costs per mirror square meter is in the range of 10-25%. Also, it is concluded that linear Fresnel collectors are more suitable for low latitude areas. As a consequence of the studies carried out in this thesis, an innovative storage system has been patented. This system takes into account the variation of the flux intensity along the day, especially for East-West oriented solar fields. As a result, the invention would allow to exploit the impinging radiation along longer time every day, increasing appreciably the optical and thermal efficiencies.

Optimized square Fresnel zone plates for microoptics applications

Polygonal Fresnel zone plates with a low number of sides have deserved attention in micro and nanoptics, because they can be straightforwardly integrated in photonic devices, and, at the same time, they represent a balance between the high-focusing performance of a circular zone plate and the easiness of fabrication at micro and nano-scales of polygons. Among them, the most representative family are Square Fresnel Zone Plates (SFZP). In this work, we propose two different customized designs of SFZP for optical wavelengths. Both designs are based on the optimization of a SFZP to perform as close as possible as a usual Fresnel Zone Plate. In the first case, the criterion followed to compute it is the minimization of the difference between the area covered by the angular sector of the zone of the corresponding circular plate and the one covered by the polygon traced on the former. Such a requirement leads to a customized polygon-like Fresnel zone. The simplest one is a square zone with a pattern of phases repeating each five zones. On the other hand, an alternative SFZP can be designed guided by the same criterion but with a new restriction. In this case, the distance between the borders of different zones remains unaltered. A comparison between the two lenses is carried out. The irradiance at focus is computed for both and suitable merit figures are defined to account for the difference between them.

Optimal phase distributions for polygonal Fresnel lenses

Polygonal Fresnel zone plates can be configured in a variety of forms depending on the number of sides of the polygon and the number of phase steps used. This contribution deals with some specific polygonal designs that tessellate the plane: triangles, squares, and hexagons. The phase distribution is chosen as a continuous one to form a polygonal kinoform. The selected designs have been simulated and its behaviour compared. Although their performance is worse than the circular Fresnel plate, they may present some other advantages as the tessellation capability, and the possibility to fabricate them as extruded profiles.

Petroleomics By Electrospray Ionization Ft-icr Mass Spectrometry Coupled To Partial Least Squares With Variable Selection Methods: Prediction Of The Total Acid Number Of Crude Oils.

Negative-ion mode electrospray ionization, ESI(-), with Fourier transform ion cyclotron resonance mass spectrometry (FT-ICR MS) was coupled to a Partial Least Squares (PLS) regression and variable selection methods to estimate the total acid number (TAN) of Brazilian crude oil samples. Generally, ESI(-)-FT-ICR mass spectra present a power of resolution of ca. 500,000 and a mass accuracy less than 1 ppm, producing a data matrix containing over 5700 variables per sample. These variables correspond to heteroatom-containing species detected as deprotonated molecules, [M - H](-) ions, which are identified primarily as naphthenic acids, phenols and carbazole analog species. The TAN values for all samples ranged from 0.06 to 3.61 mg of KOH g(-1). To facilitate the spectral interpretation, three methods of variable selection were studied: variable importance in the projection (VIP), interval partial least squares (iPLS) and elimination of uninformative variables (UVE). The UVE method seems to be more appropriate for selecting important variables, reducing the dimension of the variables to 183 and producing a root mean square error of prediction of 0.32 mg of KOH g(-1). By reducing the size of the data, it was possible to relate the selected variables with their corresponding molecular formulas, thus identifying the main chemical species responsible for the TAN values.

The Basic Reproduction Number Obtained From Jacobian And Next Generation Matrices - A Case Study Of Dengue Transmission Modelling.

The basic reproduction number is a key parameter in mathematical modelling of transmissible diseases. From the stability analysis of the disease free equilibrium, by applying Routh-Hurwitz criteria, a threshold is obtained, which is called the basic reproduction number. However, the application of spectral radius theory on the next generation matrix provides a different expression for the basic reproduction number, that is, the square root of the previously found formula. If the spectral radius of the next generation matrix is defined as the geometric mean of partial reproduction numbers, however the product of these partial numbers is the basic reproduction number, then both methods provide the same expression. In order to show this statement, dengue transmission modelling incorporating or not the transovarian transmission is considered as a case study. Also tuberculosis transmission and sexually transmitted infection modellings are taken as further examples.

Genomic Copy Number Alterations Of Primary And Secondary Metastasizing Pleomorphic Adenomas.

Metastasizing pleomorphic adenoma (MPA) is a rare tumour, and its mechanism of metastasis still is unknown. To date, there has been no study on MPA genomics. We analysed primary and secondary MPAs with array comparative genomic hybridization to identify somatic copy number alterations and affected genes. Tumour DNA samples from primary (parotid salivary gland) and secondary (scalp skin) MPAs were subjected to array comparative genomic hybridization investigation, and the data were analysed with NEXUS COPY NUMBER DISCOVERY. The primary MPA showed copy number losses affecting 3p22.2p14.3 and 19p13.3p123, and a complex pattern of four different deletions at chromosome 6. The 3p deletion encompassed several genes: CTNNB1, SETD2, BAP1, and PBRM1, among others. The secondary MPA showed a genomic profile similar to that of the primary MPA, with acquisition of additional copy number changes affecting 9p24.3p13.1 (loss), 19q11q13.43 (gain), and 22q11.1q13.33 (gain). Our findings indicated a clonal origin of the secondary MPA, as both tumours shared a common profile of genomic copy number alterations. Furthermore, we were able to detect in the primary tumour a specific pattern of copy number alterations that could explain the metastasizing characteristic, whereas the secondary MPA showed a more unbalanced genome.

Investigation Of Genetic Factors Underlying Typical Orofacial Clefts: Mutational Screening And Copy Number Variation.

Typical orofacial clefts (OFCs) comprise cleft lip, cleft palate and cleft lip and palate. The complex etiology has been postulated to involve chromosome rearrangements, gene mutations and environmental factors. A group of genes including IRF6, FOXE1, GLI2, MSX2, SKI, SATB2, MSX1 and FGF has been implicated in the etiology of OFCs. Recently, the role of the copy number variations (CNVs) has been studied in genetic defects and diseases. CNVs act by modifying gene expression, disrupting gene sequence or altering gene dosage. The aims of this study were to screen the above-mentioned genes and to investigate CNVs in patients with OFCs. The sample was composed of 23 unrelated individuals who were grouped according to phenotype (associated with other anomalies or isolated) and familial recurrence. New sequence variants in GLI2, MSX1 and FGF8 were detected in patients, but not in their parents, as well as in 200 control chromosomes, indicating that these were rare variants. CNV screening identified new genes that can influence OFC pathogenesis, particularly highlighting TCEB3 and KIF7, that could be further analyzed. The findings of the present study suggest that the mechanism underlying CNV associated with sequence variants may play a role in the etiology of OFC.

Malpighiaceae: correlations between habit, fruit type and basic chromosome number

The family Malpighiaceae presents species with different habits, fruit types and cytological characters. Climbers are considered the most derived habit, followed, respectively, by the shrubby and arboreal ones. The present study examines the relationship between basic chromosome numbers and the derivation of climbing habit and fruit types in Malpighiaceae. A comparison of all the chromosome number reports for Malpighiaceae showed a predominance of chromosome numbers based on x=5 or 10 in the genera of sub-family Malpighioideae, mainly represented by climbers with winged fruits, whereas non-climbing species with non-winged fruits, which predominate in sub-family Byrsonimoideae, had counts based on x=6, which is considered the less derived basic number for the family. Based on such data, confirmed by statistic assays, and on the monophyletic origin of this family, we admit the hypothesis that morphological derivation of habit and fruit is correlated with chromosome basic number variation in the family Malpighiaceae.

Influence of English language in the number of citations of articles published in Brazilian journals of Ophthalmology

PURPOSE: To determine the association between language and number of citations of ophthalmology articles published in Brazilian journals. METHODS: This study was a systematic review. Original articles were identified by review of documents published at the two Brazilian ophthalmology journals indexed at Science Citation Index Expanded - SCIE [Arquivos Brasileiros de Oftalmologia (ABO) and Revista Brasileira de Oftalmologia (RBO)]. All document types (articles and reviews) listed at SCIE in English (English Group) or in Portuguese (Portuguese Group) from January 1, 2008 to December 31, 2009 were included, except: editorial materials; corrections; letters; and biographical items. The primary outcome was the number of citations through the end of second year after publication date. Subgroup analysis included likelihood of citation (cited at least once versus no citation), journal, and year of publication. RESULTS: The search at the web of science revealed 382 articles [107 (28%) in the English Group and 275 (72%) in the Portuguese Group]. Of those, 297 (77.7%) were published at the ABO and 85 (23.3%) at the RBO. The citation counts were statistically significantly higher (P<0.001) in the English Group (1.51 - SD 1.98 - range 0 to 11) compared with the Portuguese Group (0.57 - SD 1.06 - range 0 to 7). The likelihood citation was statistically significant higher (P<0.001) in the English Group (70/107 - 65.4%) compared with the Portuguese Group (89/275 - 32.7%). There were more articles published in English at the ABO (98/297 - 32.9%) than at the RBO (9/85 - 10.6%) [P<0.001]. There were no significant difference (P=0.967) at the proportion of articles published in English at the years 2008 (48/172 - 27.9%) and 2009 (59/210 - 28.1%). CONCLUSION: The number of citations of articles published in Portuguese at Brazilian ophthalmology journals is lower than the published in English. The results of this study suggest that the editorial boards should strongly encourage the authors to adopt English as the main language in their future articles.

Relationship among malocclusion, number of occlusal pairs and mastication

This study evaluated the relationship among malocclusion, number of occlusal pairs, masticatory performance, masticatory time and masticatory ability in completely dentate subjects. Eighty healthy subjects (mean age = 19.40 ± 4.14 years) were grouped according to malocclusion diagnosis (n = 16): Class I, Class Class II-2, Class III and Normocclusion (control). Number of occlusal pairs was determined clinically. Masticatory performance was evaluated by the sieving method, and the time used for the comminute test food was registered as the masticatory time. Masticatory ability was measured by a dichotomic self-perception questionnaire. Statistical analysis was done by one-way ANOVA, ANOVA on ranks, Chi-Square and Spearman tests. Class II-1 and III malocclusion groups presented a smaller number of occlusal pairs than Normocclusion (p < 0.0001), Class I (p < 0.001) and II-2 (p < 0.0001) malocclusion groups. Class I, and III malocclusion groups showed lower masticatory performance values compared to Normocclusion (p < 0.05) and Class II-2 (p < 0.05) malocclusion groups. There were no differences in masticatory time (p = 0.156) and ability (χ2 = 3.58/p= 0.465) among groups. Occlusal pairs were associated with malocclusion (rho = 0.444/p < 0.0001) and masticatory performance (rho = 0.393/p < 0.0001), but malocclusion was not correlated with masticatory performance (rho = 0.116/p= 0.306). In conclusion, masticatory performance and ability were not related to malocclusion, and subjects with Class I, II-1 and III malocclusions presented lower masticatory performance because of their smaller number of occlusal pairs.

An analytical solution for the critical number of particles for stable bose-einstein condensation under the influence of an anisotropic potential

We have considered a Bose gas in an anisotropic potential. Applying the the Gross-Pitaevskii Equation (GPE) for a confined dilute atomic gas, we have used the methods of optimized perturbation theory and self-similar root approximants, to obtain an analytical formula for the critical number of particles as a function of the anisotropy parameter for the potential. The spectrum of the GPE is also discussed.

Within- and between-person variations as determinant factors to calculate the number of observations to estimate usual dietary intake of adolescents

This cross-sectional study describes the variation in the dietary intake of energy and macronutrients based on estimated coefficients of within- and between-person variation and intends to calculate the number of days required to evaluate the dietary intake of adolescents in Piracicaba, Brazil. The sample was made up by ninety-two adolescents aged between eleven and sixteen years. Interviews were performed to assess demographic, anthropometric and dietary variables, which were evaluated by the answers obtained through two 24-hour recalls. Descriptive analyses were performed and variances and Coefficients of variation within- and between-person were obtained by results of one-way ANOVA. The mean energy intake observed was 2,326.2 kilocalories; mean macronutrient intake came to 89.0 grams of fatty acids, 305.3 grams of carbohydrates and 82.2 grams of proteins. Coefficients of within-person variation ranged from 36% to 45%, while between-person coefficients varied from 26% to 31%. Variance ratios ranged from 1.35 (carbohydrate) to 2.62 (protein). The lowest number of observations required to correctly evaluate the usual diet calculated (r = 0.90) was six days, for carbohydrates while the highest was eleven, for proteins. Coefficients of variation were similar to those observed in international studies. We conclude that, in this study, two observations were sufficient to obtain reasonable correlations. However, six evaluations are necessary for these adolescents in order to obtain excellent correlations.