The VLT-FLAMES survey of massive stars: atmospheric parameters and rotational velocity distributions for B-type stars in the Magellanic Clouds

Aims.We aim to provide the atmospheric parameters and rotational velocities for a large sample of O- and early B-type stars, analysed in a homogeneous and consistent manner, for use in constraining theoretical models. Methods: Atmospheric parameters, stellar masses, and rotational velocities have been estimated for approximately 250 early B-type stars in the Large (LMC) and Small (SMC) Magellanic Clouds from high-resolution VLT-FLAMES data using the non-LTE TLUSTY model atmosphere code. This data set has been supplemented with our previous analyses of some 50 O-type stars (Mokiem et al. 2006, 2007) and 100 narrow-lined early B-type stars (Hunter et al. 2006; Trundle et al. 2007) from the same survey, providing a sample of ~400 early-type objects. Results: Comparison of the rotational velocities with evolutionary tracks suggests that the end of core hydrogen burning occurs later than currently predicted and we argue for an extension of the evolutionary tracks. We also show that the large number of the luminous blue supergiants observed in the fields are unlikely to have directly evolved from main-sequence massive O-type stars as neither their low rotational velocities nor their position on the H-R diagram are predicted. We suggest that blue loops or mass-transfer binary systems may populate the blue supergiant regime. By comparing the rotational velocity distributions of the Magellanic Cloud stars to a similar Galactic sample, we find that (at 3s confidence level) massive stars (above 8 M?) in the SMC rotate faster than those in the solar neighbourhood. However there appears to be no significant difference between the rotational velocity distributions in the Galaxy and the LMC. We find that the v sin i distributions in the SMC and LMC can modelled with an intrinsic rotational velocity distribution that is a Gaussian peaking at 175 km s-1 (SMC) and 100 km s-1 (LMC) with a 1/e half width of 150 km s-1. We find that in NGC 346 in the SMC, the 10-25 M? main-sequence stars appear to rotate faster than their higher mass counterparts. It is not expected that O-type stars spin down significantly through angular momentum loss via stellar winds at SMC metallicity, hence this could be a reflection of mass dependent birth spin rates. Recently Yoon et al. (2006) have determined rates of GRBs by modelling rapidly rotating massive star progenitors. Our measured rotational velocity distribution for the 10-25 M? stars is peaked at slightly higher velocities than they assume, supporting the idea that GRBs could come from rapid rotators with initial masses as low as 14 M? at low metallicities.

Extraction and purification of triterpenic acids from eucalypt bark

A indústria da pasta e do papel é um sector importante da economia mundial, particularmente a que assenta em espécies de Eucalyptus. No entanto, essas indústrias geram quantidades significativas de correntes secundárias de subprodutos e resíduos de biomassa que podem ser exploradas em aplicações de valor acrescentado em vez de serem queimadas para produção de energia. Um exemplo nobre pode ser a produção de ácidos triterpénicos com estruturas dos tipos lupano, oleanano e ursano, dada a sua abundância em alguns destes resíduos, em particular na casca, adotando o conceito de biorrefinaria integrada numa fábrica de pasta. Estes compostos são conhecidos pelas suas inúmeras actividades biológicas, por exemplo, antioxidante, anti-inflamatória e antitumoral, e podem encontrar aplicações em produtos de elevado valor, tais como cosméticos, nutracêuticos ou farmacêuticos. Neste sentido, o estudo das potencialidades das cascas das espécies de eucalipto mais exploradas enquanto fontes de compostos triterpénicos é um tópico relevante. Por conseguinte, foram analisados e comparados em pormenor os teores e composições em ácidos triterpénicos (TTAs) das cascas externas de várias espécies de eucalipto (E. globulus, E. grandis, E. urograndis, E. maidenii e E. nitens). Os teores dos principais TTAs identificados nestas espécies variaram entre 4.5 g/kg no E. urograndis e 21.6 g/kg no E. nitens. Observou-se que as cascas externas de Eucalyptus de zonas temperadas e Mediterrânicas, nomeadamente E. nitens e E. globulus, são mais ricas em TTAs que as espécies de regiões tropicais e subtropicais. Além disso, a casca externa do E. globulus é claramente a mais rica em ácidos com estruturas do tipo ursano enquanto a do E. nitens é a mais rica em ácidos do tipo oleanano e lupano. Estes resultados levaram-nos a estudar a extração dos TTAs da casca de Eucalyptus, bem como a sua posterior concentração e purificação, a qual foi efetuada por extração sólido-líquido convencional combinada com a precipitação de solutos, e por extração com fluidos supercríticos (SFE). No que diz respeito à primeira abordagem referida, foi desenvolvido neste trabalho um método patenteado que permite obter extratos enriquecidos em TTAs das cascas de eucalipto baseado em tecnologias disponíveis no imediato. Em relação à segunda abordagem, e de forma a apostar em processos de baixo impacto ambiental exigidos pelas biorrefinarias do futuro, a SFE surge como uma opção natural. Assim, foi efetuada a SFE da casca caduca do E. globulus com dióxido de carbono puro e modificado para recuperar a fração de TTAs, e os resultados foram comparados com os obtidos por extração em Soxhlet com diclorometano. Foram realizados estudos preliminares sobre a influência da pressão (100-200 bar), a adição de co-solvente (0, 5 e 8% m/m de etanol), e operação em múltiplos passos a fim de avaliar a aplicabilidade da alternativa supercrítica para a sua produção eficiente e selectiva. Os resultados destacaram a influência da pressão e o importante papel resumo (cont.) desempenhado pelo co-solvente neste processo, cujo efeito foi mais relevante do que o aumento da pressão em várias dezenas de bar. Este trabalho foi depois otimizado, usando o planeamento factorial de experiências e a metodologia de superfícies de resposta, para analisar a influência da temperatura (40-60 ºC), pressão (100-200 bar), e teor de etanol (0.0-5.0% m/m) na recuperação dos TTAs e respectiva concentração nos extractos. Nestes intervalos, as melhores condições de operação encontradas foram 200 bar, 40 °C e 5% de etanol, para as quais os modelos de regressão estatisticamente validados previram um rendimento de extração de 1.2% com 50% de concentração em TTAs, correspondendo ao rendimento em TTAs de 5.1 g/kg de casca e uma recuperação de 79.2% comparativamente ao valor do Soxhlet. Os TTAs livres e acetilados apresentaram tendências de extracção bastante distintas devido às suas diferentes afinidades para o CO2 causadas pelas diferentes polaridades: os derivados acetilados aproximam-se de um patamar máximo a cerca de 200 bar e 5% de etanol, enquanto a extração dos TTAs livres apresenta uma tendência sempre crescente no intervalo de condições estudado. Foram também medidas curvas cumulativas de SFE da casca do E. globulus de forma a analisar o comportamento cinético do processo em termos de rendimento total, rendimento em TTAs, rendimento em TTAs livres, rendimento em TTAs acetilados, e concentração dos TTAs nos extractos. Foi analisada a influência da pressão, temperatura, teor de co-solvente e caudal do dióxido de carbono sobre as respostas anteriores. Os dados experimentais foram modelados com os modelos Logístico, de Dessorção, de Placa Plana Simples, e de Difusão. Na globalidade, os resultados confirmaram que a pressão e o teor de etanol têm um efeito significativo sobre as curvas de extração, os rendimentos finais e as concentrações dos extratos, e mostraram a presença de limitações externas à transferência de massa em alguns ensaios. Mais uma vez, as famílias individuais de TTAs livres e acetilados apresentaram diferentes tendências de extracção. A modelação permitiu-nos confirmar não só o importante papel desempenhado pela difusão intraparticular na SFE, mas também a contribuição da resistência no filme em alguns ensaios. Após a análise de todos os resultados, foi efetuado um ensaio em duas etapas em série, possibilitando o enriquecimento do teor em TTAs no extracto devido às diferentes condições adotadas em cada etapa. Por último, um éster metílico de um ácido triterpénico do tipo oleanano - morolato de metilo - foi identificado pela primeira vez enquanto componente da casca de Eucalyptus na casca externa do Eucalyptus grandis x globulus, onde ocorre em teores elevados. A sua extração com CO2 supercrítico foi também realizada, visando a conceção de uma alternativa de extração ambientalmente benigna para este composto. A 200 bar e 60 ºC, a remoção do morolato de metilo atingiu um patamar às 6 h para 5.1 kg h-1 de CO2 / kg de casca. Em geral, e de forma semelhante à SFE da casca do E. globulus, os TTAs acetilados foram mais significativamente extraídos quando comparados com os seus ácidos livres, o que está diretamente relacionado com a natureza menos polar destas moléculas. O trabalho apresentado nesta tese é uma contribuição para a valorização de uma corrente de biomassa com baixo valor na indústria de pasta em duas vertentes complementares. Por um lado, aumentou o conhecimento da composição lipofílica das cascas de Eucalyptus spp. com interesse comercial para a produção de pasta, destacando o seu potencial enquanto fontes de ácidos triterpénicos. Por outro lado, foram desenvolvidos dois processos alternativos e facilmente integráveis numa fábrica de pasta para a sua exploração a partir da casca: um baseado em tecnologias convencionais bem estabelecidas a nível industrial, prevendo a sua aplicação a curto prazo, e um outro baseado na SFE, seguindo as tendências das futuras biorrefinarias.

Mathematical modelling of supercritical CO2 extraction of volatile oils from aromatic plants

The modelling of the experimental data of the extraction of the volatile oil from six aromatic plants (coriander, fennel, savoury, winter savoury, cotton lavender and thyme) was performed using five mathematical models, based on differential mass balances. In all cases the extraction was internal diffusion controlled and the internal mass transfer coefficienty (k(s)) have been found to change with pressure, temperature and particle size. For fennel, savoury and cotton lavender, the external mass transfer and the equilibrium phase also influenced the second extraction period, since k(s) changed with the tested flow rates. In general, the axial dispersion coefficient could be neglected for the conditions studied, since Peclet numbers were high. On the other hand, the solute-matrix interaction had to be considered in order to ensure a satisfactory description of the experimental data.

Heat transfer coefficients from Newtonian and non-Newtonian fluids flowing in laminar regime in a helical coil

This study aimed to carry out experimental work to obtain, for Newtonian and non-Newtonian fluids, heat transfer coefficients, at constant wall temperature as boundary condition, in fully developed laminar flow inside a helical coil. The Newtonian fluids studied were aqueous solutions of glycerol, 25%, 36%, 43%, 59% and 78% (w/w) and the non-Newtonian fluids aqueous solutions of carboxymethylcellulose (CMC), a polymer, with concentrations 0.1%, 0.2%, 0.3%, 0.4% and 0.6% (w/w) and aqueous solutions of xanthan gum (XG), another polymer, with concentrations 0.1% and 0.2% (w/w). According to the rheological study performed, the polymer solutions had shear thinning behavior and different values of elasticity. The helical coil used has internal diameter, curvature ratio, length and pitch, respectively: 0.004575 m, 0.0263, 5.0 m and 11.34 mm. The Nusselt numbers for the CMC solutions are, on average, slightly higher than those for Newtonian fluids, for identical Prandtl and generalized Dean numbers. As outcome, the viscous component of the shear thinning polymer tends to potentiate the mixing effect of the Dean cells. The Nusselt numbers of the XG solutions are significant lower than those of the Newtonian solutions, for identical Prandtl and generalized Dean numbers. Therefore, the elastic component of the polymer tends to diminish the mixing effect of the Dean cells. A global correlation, for Nusselt number as a function of Péclet, generalized Dean and Weissenberg numbers for all Newtonian and non-Newtonian solutions studied, is presented.

Project, modelling and simulation of air stripping columns for the treatment of groundwater contaminated with volatile organic compounds

Volatile organic compounds are a common source of groundwater contamination that can be easily removed by air stripping in columns with random packing and using a counter-current flow between the phases. This work proposes a new methodology for the column design for any particular type of packing and contaminant avoiding the necessity of a pre-defined diameter used in the classical approach. It also renders unnecessary the employment of the graphical Eckert generalized correlation for pressure drop estimates. The hydraulic features are previously chosen as a project criterion and only afterwards the mass transfer phenomena are incorporated, in opposition to conventional approach. The design procedure was translated into a convenient algorithm using C++ as programming language. A column was built in order to test the models used either in the design or in the simulation of the column performance. The experiments were fulfilled using a solution of chloroform in distilled water. Another model was built to simulate the operational performance of the column, both in steady state and in transient conditions. It consists in a system of two partial non linear differential equations (distributed parameters). Nevertheless, when flows are steady, the system became linear, although there is not an evident solution in analytical terms. In steady state the resulting system of ODE can be solved, allowing for the calculation of the concentration profile in both phases inside the column. In transient state the system of PDE was numerically solved by finite differences, after a previous linearization.

Practical software improve design and performance of a randomly packed air stripping column for removing volatile organic compounds from water

STRIPPING is a software application developed for the automatic design of a randomly packing column where the transfer of volatile organic compounds (VOCs) from water to air can be performed and to simulate it’s behaviour in a steady-state. This software completely purges any need of experimental work for the selection of diameter of the column, and allows a choice, a priori, of the most convenient hydraulic regime for this type of operation. It also allows the operator to choose the model used for the calculation of some parameters, namely between the Eckert/Robbins model and the Billet model for estimating the pressure drop of the gaseous phase, and between the Billet and Onda/Djebbar’s models for the mass transfer. Illustrations of the graphical interface offered are presented.

Impacts de l'écoulement souterrain sur la dégradation du pergélisol

Les changements climatiques mesurés dans le Nord-ouest canadien au cours du XXIe siècle entraînent une dégradation du pergélisol. Certaines des principales conséquences physiques sont la fonte de la glace interstitielle lors du dégel du pergélisol, l’affaissement du sol et la réorganisation des réseaux de drainage. L’effet est particulièrement marqué pour les routes bâties sur le pergélisol, où des dépressions et des fentes se créent de façon récurrente, rendant la conduite dangereuse. Des observations et mesures de terrain effectuées à Beaver Creek (Yukon) entre 2008 et 2011 ont démontré qu’un autre processus très peu étudié et quantifié dégradait le pergélisol de façon rapide, soit la chaleur transmise au pergélisol par l’écoulement souterrain. Suite aux mesures de terrain effectuées (relevé topographique, étude géotechnique du sol, détermination de la hauteur de la nappe phréatique et des chenaux d’écoulement préférentiels, température de l’eau et du sol, profondeur du pergélisol et de la couche active), des modèles de transfert de chaleur par conduction et par advection ont été produits. Les résultats démontrent que l’écoulement souterrain dans la couche active et les zones de talik contribue à la détérioration du pergélisol via différents processus de transfert de chaleur conducto-convectifs. L’écoulement souterrain devrait être pris en considération dans tous les modèles et scénarios de dégradation du pergélisol. Avec une bonne caractérisation de l’environnement, le modèle de transfert de chaleur élaboré au cours de la présente recherche est applicable dans d’autres zones de pergélisol discontinu.

Aprofitament de residus vegetals per a la concentració i separació d'ions metàl·lics de solucions aquoses

La adsorció sobre carbó actiu és un tractament habitual para a l'eliminació de metalls contaminants d'afluents aquosos. Amb la finalitat de reduir el cost dels tractament actuals s'està estudiant la utilització d'adsorbents de baix cost com a substituts dels sorbents convencionals que s'utilitzen usualment en els tractaments d'aigües. En aquesta tesi s'ha avaluat la possibilitat d'aprofitar diversos residus vegetals procedents de processos agrícoles o residus industrials, com a sorbents per a l'eliminació de metalls de solucions aquoses. En concret, el treball s'ha centrat en assolir dos grans objectius, primer de tot, conèixer la possibilitat d'utilitzar el pinyol d'oliva com a adsorbent de metalls divalents de solucions aquoses, com són el Cd(II), Cu(II), Ni(II) i Pb(II), i en segon lloc, avaluar la utilització de quatre residus vegetals (escorça de yohimbe, rapa de raïm, suro i pinyol d'oliva) per a l'extracció de Cr(VI) de solucions aquoses i utilitzar el residu vegetal més apropiat com a base per tal d'obtenir un nou sorbent, amb més bona morfologia per ser utilitzat en sistemes d'eliminació de Cr(VI) en continu. Les dades cinètiques d'adsorció es van tractar amb el model cinètic de pseudo-segon ordre, amb el que es van determinar la velocitat inicial d'adsorció dels metalls. Els models d'isotermes d'equilibri de Langmuir i Freundlich es van utilitzar per descriure els equilibris d'adsorció i calcular els paràmetres de les isotermes. En els assajos amb mescles binàries de metalls, els resultats es van tractar amb el model ampliat de Langmuir. En aquest treball s'ha determinat, primerament, que el pinyol d'oliva és un bon sorbent dels metalls divalents estudiats. S'ha determinat que l'adsorció dels metalls varia amb el pH de la solució i que la presència de sals en la solució provoca una davallada en l'adsorció. També es va determinar la competència en l'adsorció entre els metalls al realitzar assajos en mescles binàries. Amb els estudis d'extracció de Cr(VI) amb els quatre materials s'ha determinat que aquests vegetals poden adsorbir el Cr(VI) en solució, tot i que la rapa i la yohimbe són els materials que presenten unes capacitats d'adsorció més elevades. Les anàlisis amb espectroscòpia electrònica de Raig-X realitzats en la rapa de raïm i la yohimbe en contacte amb Cr(VI) varen demostrar la presència del metall en els dos estats d'oxidació, Cr(VI) i Cr(III), evidenciant la participació del mecanisme de reducció de Cr(VI) en l'adsorció del metall sobre els materials. Amb l'encapsulació de la rapa de raïm es va incrementar significativament la capacitat d'adsorció del metall respecta a la seva utilització en la forma original. Les dades experimentals es van tractar amb un model de difusió i es va determinar el coeficient de difusió en l'adsorció en el sorbent amb diferent percentatge de rapa en l'interior. Finalment s'han realitzat estudis preliminars d'adsorció de crom hexavalent en continu utilitzant columnes de petites dimensions farcides amb el sorbent format per boles de rapa encapsulada en alginat càlcic. En aquests assajos s'ha determinat l'efecte en l'adsorció de metall de diversos paràmetres d'operació, com la concentració inicial de metall, el cabal i l'alçada de llit. El model de transferència de matèria i difusió en columna emprat pel tractament de les dades experimentals va mostrar l'efecte de la concentració inicial i del cabal en els coeficients de difusió i transferència de matèria. El model BDST no va predir correctament el temps de servei en els diferents paràmetres d'operació experimentats a causa de la complexitat del procés d'adsorció. Amb aquests assajos preliminars s'han establert les bases de l'adsorció de Cr(VI) amb el nou sorbent en sistemes en continu amb la finalitat de realitzar estudis posteriors d'eliminació del metall en columnes de majors dimensions.

Theoretical and experimental approaches towards the determination of solute effective diffusivities in foods

The aim of this review paper is to present experimental methodologies and the mathematical approaches used to determine effective diffusivities of solutes in food materials. The paper commences by describing the diffusion phenomena related to solute mass transfer in foods and effective diffusivities. It then focuses on the mathematical formulation for the calculation of effective diffusivities considering different diffusion models based on Fick's second law of diffusion. Finally, experimental considerations for effective diffusivity determination are elucidated primarily based on the acquirement of a series of solute content versus time curves appropriate to the equation model chosen. Different factors contributing to the determination of the effective diffusivities such as the structure of food material, temperature, diffusion solvent, agitation, sampling, concentration and different techniques used are considered. (c) 2005 Elsevier Inc. All rights reserved.

Spectrally invariant approximation within atmospheric radiative transfer

Certain algebraic combinations of single scattering albedo and solar radiation reflected from, or transmitted through, vegetation canopies do not vary with wavelength. These ‘‘spectrally invariant relationships’’ are the consequence of wavelength independence of the extinction coefficient and scattering phase function in veg- etation. In general, this wavelength independence does not hold in the atmosphere, but in cloud-dominated atmospheres the total extinction and total scattering phase function vary only weakly with wavelength. This paper identifies the atmospheric conditions under which the spectrally invariant approximation can accu- rately describe the extinction and scattering properties of cloudy atmospheres. The validity of the as- sumptions and the accuracy of the approximation are tested with 1D radiative transfer calculations using publicly available radiative transfer models: Discrete Ordinate Radiative Transfer (DISORT) and Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART). It is shown for cloudy atmospheres with cloud optical depth above 3, and for spectral intervals that exclude strong water vapor absorption, that the spectrally invariant relationships found in vegetation canopy radiative transfer are valid to better than 5%. The physics behind this phenomenon, its mathematical basis, and possible applications to remote sensing and climate are discussed.

Transit times – the link between hydrology and water quality at the catchment scale

In spite of trying to understand processes in the same spatial domain, the catchment hydrology and water quality scientific communities are relatively disconnected and so are their respective models. This is emphasized by an inadequate representation of transport processes, in both catchment-scale hydrological and water quality models. While many hydrological models at the catchment scale only account for pressure propagation and not for mass transfer, catchment scale water quality models are typically limited by overly simplistic representations of flow processes. With the objective of raising awareness for this issue and outlining potential ways forward we provide a non-technical overview of (1) the importance of hydrology-controlled transport through catchment systems as the link between hydrology and water quality; (2) the limitations of current generation catchment-scale hydrological and water quality models; (3) the concept of transit times as tools to quantify transport and (4) the benefits of transit time based formulations of solute transport for catchment-scale hydrological and water quality models. There is emerging evidence that an explicit formulation of transport processes, based on the concept of transit times has the potential to improve the understanding of the integrated system dynamics of catchments and to provide a stronger link between catchment-scale hydrological and water quality models.

A singular integro-differential equation model for dryout in LMFBR boiler tubes

A 2D steady model for the annular two-phase flow of water and steam in the steam-generating boiler pipes of a liquid metal fast breeder reactor is proposed The model is based on thin-layer lubrication theory and thin aerofoil theory. The exchange of mass between the vapour core and the liquid film due to evaporation of the liquid film is accounted for using some simple thermodynamics models, and the resultant change of phase is modelled by proposing a suitable Stefan problem Appropriate boundary conditions for the now are discussed The resulting non-lineal singular integro-differential equation for the shape of the liquid film free surface is solved both asymptotically and numerically (using some regularization techniques) Predictions for the length to the dryout point from the entry of the annular regime are made The influence of both the traction tau provided by the fast-flowing vapour core on the liquid layer and the mass transfer parameter eta on the dryout length is investigated

Aproveitamento de subprodutos da industrialização do escargot

The waste in the industries of escargot processing is very big. This is composed basically of escargot meat out of the commercialization patterns and the visceras. In this context, there is a need to take advantage to the use of these sub-products. A possibility should be drying them and transforming them in a certain form to be reused. Than, the present work has the objective of studying the reutilization of the sub-products of the escargot industrialization for by means of drying process. The samples were transformed in pastes, through a domestic processor for approximately 1 minute and compacted in trays of aluminum without perforations with three different heights (5 mm, 10 mm and 15 mm). The drying was accomplished in a tray dryer with air circulation and transverse flow at a speed of 0,2 m/s and three temperature levels (70°C, 80°C and 90ºC). A drying kinetics study was accomplished for the obtained curves and for the heat and mass transfer coefficients using experimental procedures based in an experimental planning of 22 factorial type. Microbiological and physiochemical analysis were also accomplished for the in nature and the dehydrated sub-products. In the drying process, it was observed the great importance of the external resistances to the mass transfer and heat in the period of constant tax influenced by the temperature. The evaporation taxes indicated a mixed control of the mass transfer for the case of the thickest layers. As already expected, the drying constant behavior was influenced by the temperature and thickness of the medium, increasing and decreasing. The statistical analysis of the results, in agreement with the factorial planning 22, showed that the fissures, the shrinking of the transfer area and the formation of a crust on the surface might have contributed to the differences between the practical results and the linear model proposed. The temperature and the thickness influenced significantly in the answers of the studied variables: evaporation tax and drying constant. They were obtained significant statistical models and predictive ones for evaporation tax for the meat as well as for the visceras

Sistema de secagem solar para frutos tropicais e modelagem da secagem de banana em um secador de coluna estática

In this study were projected, built and tested an electric solar dryer consisting of a solar collector, a drying chamber, an exhaust fan and a fan to promote forced hot air convection. Banana drying experiments were also carried out in a static column dryer to model the drying and to obtain parameters that can be used as a first approximation in the modeling of an electric solar dryer, depending on the similarity of the experimental conditions between the two drying systems. From the banana drying experiments conducted in the static column dryer, we obtained food weight data as a function of aqueous concentration and temperature. Simplified mathematical models of the banana drying were made, based on Fick s and Fourier s second equations, which were tested with the experimental data. We determined and/or modeled parameters such as banana moisture content, density, thin layer drying curves, equilibrium moisture content, molecular diffusivity of the water in banana DAB, external mass transfer coefficient kM, specific heat Cp, thermal conductivity k, latent heat of water evaporation in the food Lfood, time to heat food, and minimum energy and power required to heat the food and evaporate the water. When we considered the shrinkage of radius R of a banana, the calculated values of DAB and kM generally better represent the phenomenon of water diffusion in a solid. The latent heat of water evaporation in the food Lfood calculated by modeling is higher than the latent heat of pure water evaporation Lwater. The values calculated for DAB and KM that best represent the drying were obtained with the analytical model of the present paper. These values had good agreement with those assessed with a numeric model described in the literature, in which convective boundary condition and food shrinkage are considered. Using parameters such as Cp, DAB, k, kM and Lfood, one can elaborate the preliminary dryer project and calculate the economy using only solar energy rather than using solar energy along with electrical energy