Simultaneous multielement extraction with the Mehlich-1 solution for Southern Brazilian soils determined by ICP-OES and the effects on the nutrients recommendations to crops

The amounts of macro (P, K, Ca and Mg) and micronutrients (Cu and Zn) extracted with the Mehlich-1 (M1) solution, by the 1.0 mol L-1 KCl (KCl) and with the 0.1 mol L-1 HCl (HCl) for representative soil types of the Rio Grande do Sul state (Brazil) were compared with those extracted with the Mehlich-1 solution determined with the inductively coupled plasma optical emission spectroscopy (ICP). The amounts of nutrients extracted by the different methods showed high correlation coefficients. On average, the Mehlich-1 solution extracted similar amounts of P, determined with colorimetric and ICP methods, and, K determined with emission and ICP. The amounts of Ca and Mg extracted with the Mehlich-1 solution, determined by ICP, were similar to those extracted with the KCl solution determined by the atomic absorption spectrophotometry. The amounts of Cu and Zn extracted with the Mehlich-1 solution, determined by the ICP, were higher than those extracted with the 0.1 mol L-1 HCl determined by the atomic absorption spectrophotometry. The results indicate that the Mehlich-1 solution and ICP can be used for simultaneous multielement extraction and determination for Southern Brazilian soils. However, a conversion factor for values interpretation is needed. The use of the conversion factor to determine the K availability index in soils is adequate and does not affect the K recommendations for crops in southern Brazilian soils.

Flooded rice yield as affected by levels of water salinity in different stages of its cycle

Losses of productivity of flooded rice in the State of Rio Grande do Sul, Brazil, may occur in the Coastal Plains and in the Southern region due to the use of saline water from coastal rivers, ponds and the Laguna dos Patos lagoon, and the sensibility of the plants are variable according to its stage of development. The purpose of this research was to evaluate the production of rice grains and its components, spikelet sterility and the phenological development of rice at different levels of salinity in different periods of its cycle. The experiment was conducted in a greenhouse, in pots filled with 11 dm³ of an Albaqualf. The levels of salinity were 0.3 (control), 0.75, 1.5, 3.0 and 4.5 dS m-1 kept in the water layer by adding a salt solution of sodium chloride, except for the control, in different periods of rice development: tillering initiation to panicle initiation; tillering initiation to full flowering; tillering initiation to physiological maturity; panicle initiation to full flowering; panicle initiation to physiological maturity and full flowering to physiological maturity. The number of panicles per pot, the number of spikelets per panicle, the 1,000-kernel weight, the spikelet sterility, the grain yield and phenology were evaluated. All characteristics were negatively affected, in a quadratic manner, with increased salinity in all periods of rice development. Among the yield components evaluated, the one most closely related to grain yields of rice was the spikelet sterility.

Nitrate and ammonium in soil solution in tobacco management systems

Tobacco farmers of southern Brazil use high levels of fertilizers, without considering soil and environmental attributes, posing great risk to water resources degradation. The objective of this study was to monitor nitrate and ammonium concentrations in the soil solution of an Entisol in and below the root zone of tobacco under conventional tillage (CT), minimum tillage (MT) and no-tillage (NT). The study was conducted in the small-watershed Arroio Lino, in Agudo, State of Rio Grande do Sul, Brazil. A base fertilization of 850 kg ha-1 of 10-18-24 and topdressing of 400 kg ha-1 of 14-0-14 NPK fertilizer were applied. The soil solution was sampled during the crop cycle with a tension lysimeter equipped with a porous ceramic cup. Ammonium and nitrate concentrations were analyzed by the distillation and titration method. Nitrate concentrations, ranging from 8 to 226 mg L-1, were highest after initial fertilization and decreased during the crop cycle. The average nitrate (N-NO3-) concentration in the root zone was 75 in NT, 95 in MT, and 49 mg L-1 in CT. Below the root zone, the average nitrate concentration was 58 under NT, 108 under MT and 36 mg L-1 under CT. The nitrate and ammonium concentrations did not differ significantly in the management systems. However, the nitrate concentrations measured represent a contamination risk to groundwater of the watershed. The ammonium concentration (N-NH4+) decreased over time in all management systems, possibly as a result of the nitrification process and root uptake of part of the ammonium by the growing plants.

Attributes of irrigated rice and soil solution as affected by salinity levels of the water layer

In the State of Rio Grande do Sul, Brazil, flooded rice fields using Patos Lagoon as the source of water for irrigation are subject to be damaged by salinity, since this source is bound to the sea on its southern end. The sensitivity of rice is variable during plant development, being higher in the seedling and reproductive periods. However, there is not enough information about the behavior of plants under salt stress during the course of its development, especially in the vegetative stage. This study evaluated the effect of different levels of salinity of irrigation water on the salinity of soil solution over time and on some plant attributes, during the vegetative stage of rice. The study was conducted in a greenhouse, where seeds of the variety IRGA 424 were sown in pots and irrigated with water with electrical conductivity (ECi) levels of: 0.3, 0.75, 1.5, 3.0 and 4.5 dS m-1; from the tillering initiation (V4) until the panicle initiation (PI). The evaluations made were the electrical conductiviy of soil solution (ECe), the dry biomass of plants and stems, tillering, height and the transpiration of plants. The ECe increased with the ECi over time, and was determined by water transpiration flux in pots. The ECe values at the end of the experiment were high and, in most cases, higher than the critical values for flooded rice. The growth attributes of rice were negatively affected from ECi of 2.0 dS m-1 and ECe of 4.0 dS m-1.

Drug-eluting beads loaded with antiangiogenic agents for chemoembolization: in vitro sunitinib loading and release and in vivo pharmacokinetics in an animal model.

PURPOSE: The combination of embolic beads with a multitargeted tyrosine kinase inhibitor that inhibits tumor vessel growth is suggested as an alternative and improvement to the current standard doxorubicin-eluting beads for use in transarterial chemoembolization. This study demonstrates the in vitro loading and release kinetics of sunitinib using commercially available embolization microspheres and evaluates the in vitro biologic efficacy on cell cultures and the resulting in vivo pharmacokinetics profiles in an animal model. MATERIALS AND METHODS: DC Bead microspheres, 70-150 µm and 100-300 µm (Biocompatibles Ltd., Farnham, United Kingdom), were loaded by immersion in sunitinib solution. Drug release was measured in saline in a USP-approved flow-through apparatus and quantified by spectrophotometry. Activity after release was confirmed in cell culture. For pharmacokinetics and in vivo toxicity evaluation, New Zealand white rabbits received sunitinib either by intraarterial injection of 100-300 µm sized beads or per os. Plasma and liver tissue drug concentrations were assessed by liquid chromatography-tandem mass spectroscopy. RESULTS: Sunitinib loading on beads was close to complete and homogeneous. A total release of 80% in saline was measured, with similar fast-release profiles for both sphere sizes. After embolization, drug plasma levels remained below the therapeutic threshold (< 50 ng/mL), but high concentrations at 6 hours (14.9 µg/g) and 24 hours (3.4 µg/g) were found in the liver tissue. CONCLUSIONS: DC Bead microspheres of two sizes were efficiently loaded with sunitinib and displayed a fast and almost complete release in saline. High liver drug concentrations and low systemic levels indicated the potential of sunitinib-eluting beads for use in embolization.

Rice grown in nutrient solution with doses of manganese and silicon

Although silicon is not recognized as a nutrient, it may benefit rice plants and may alleviate the Mn toxicity in some plant species. The dry matter yield (root, leaf, sheaths and leaf blade) and plant architecture (angle of leaf insertion and leaf arc) were evaluated in rice plants grown in nutrient solutions with three Mn doses, with and without Si addition. The treatments were arranged in a 2 x 3 factorial [with and without (2 mmol L-1) Si; three Mn doses (0.5; 2.5 and 10 µmol L-1)], in a randomized block design with 4 replications. The experimental unit was a 4 L plastic vase with 4 rice (Metica-1 cultivar) plants. Thirty nine days after keeping the seedlings in the nutrient solution the plant dry matter yield was determined; the angle of leaf insertion in the sheath and the leaf arc were measured; and the Si and Mn concentrations in roots, sheaths and leaves were determined. The analysis of variance (F test at 5 and 1 % levels) and the regression analysis (for testing plant response to Mn with the Si treatments) were performed. The Si added to the nutrient solution increased the dry matter yield of roots, sheaths and leaf blades and also decreased the angle of leaf blade insertion into the sheath and the foliar arc in the rice plant. Additionally, it ameliorated the rice plant architecture which allowed an increase in the dry matter yield. Similarly, the addition of Mn to the solution improved the architecture of the rice plants with gain in dry matter yield. As Si was added to the nutrient solution, the concentration of Mn in leaves decreased and in roots increased thus alleviating the toxic effects of Mn on the plants.

Effect of NPK fertilization on production and leaf nutrient content of eucalyptus minicuttings in nutrient solution

Adequate nutrient levels in plants vary according to the species or clone, age and management practice. Therefore, adjustments of the nutrient solution are often necessary according to the plant material for multiplication. This study aimed to evaluate the influence of NPK fertilization on production and leaf nutrient contents of eucalyptus cuttings in nutrient solution. The study was conducted from November 2008 to January 2009 in a greenhouse. The experimental design was completely randomized fractional factorial (4 x 4 x 4)½, with a total of 32 treatments with three replications. The treatments consisted of four doses of N (50, 100, 200 and 400 mg L-1) as urea, P (7.5, 15, 30 and 60 mg L-1) in the form of phosphoric acid and K (50, 100, 200 and 400 mg L-1) in the form of potassium chloride in the nutrient solution. Only the effect of N alone was significant for the number and dry weight of minicuttings per ministump, with a linear decreasing effect with increasing N levels. The highest number of cuttings was obtained at a dose of 50, 7.5 and 50 mg L-1 of N, P and K, respectively.

Zenith angle distributions at Super-Kamiokande and SNO and the solution of the solar neutrino problem

We have performed a detailed study of the zenith angle dependence of the regeneration factor and distributions of events at SNO and SK for different solutions of the solar neutrino problem. In particular, we discuss the oscillatory behavior and the synchronization effect in the distribution for the LMA solution, the parametric peak for the LOW solution, etc. A physical interpretation of the effects is given. We suggest a new binning of events which emphasizes the distinctive features of the zenith angle distributions for the different solutions. We also find the correlations between the integrated day-night asymmetry and the rates of events in different zenith angle bins. The study of these correlations strengthens the identification power of the analysis.

Deficiency symptoms and uptake of micronutrients by castor bean grown in nutrient solution

Castor bean is a nutrient-demanding species, but there is still little information on its micronutrient requirements. The objectives of this study were to evaluate the effects of levels of B (2.5, 12.5 and 25.0 µmol L-1), Cu (0.05, 0.25 and 0.50 µmol L-1), Mn (0.2, 1.0 and 2.0 µmol L-1) and Zn (0.2, 1.0 and 2.0 µmol L-1) in a nutrient solution on plant B, Cu, Mn and Zn concentrations and uptake, vegetative growth and fruit yield of castor bean "Iris", grown in greenhouse. The experiment was arranged in a completely randomized block design with three replicates. The first deficiency symptoms were observed for B, followed by Zn, Cu and Mn. The main changes in the cell ultrastructure due to lack of B were thickening of the cell walls and middle lamellae, distorted chloroplasts and tightly stacked thylakoids, besides the absence of starch grains. The Mn, Zn and Cu deficiencies led to disruption of chloroplasts, disintegration of thylakoids and absence of amyloplasts. The concentration and uptake of B, Cu, Mn, and Zn in castor bean plants increased with micronutrient supply in the solution. Fruit yield was drastically reduced by B and Mn deficiencies. On the other hand, the dry matter yield of the shoot and root of castor bean plants was not. In the treatment with full nutrient solution, the leaves accumulated 56 and 48 % of the total B and Mn taken up by the plants, respectively, and the seeds and roots 85 and 61 % of the total Cu and Zn taken up, respectively. This shows the high demand of castor bean Iris for B and Mn for fruit yield.

Dynamics of ammonium and pH in the solution of soils with different salinity levels, growing irrigated rice

Rice in Rio Grande do Sul State is grown mostly under flooding, which induces a series of chemical, physical and biological changes in the root environment. These changes, combined with the presence of rice plants, affect the availability of exchangeable ammonium (NH4+) and pH of soil solution, whereas the dynamics of both variables can be influenced by soil salinity, a common problem in the coastal region. This study was conducted to evaluate the dynamics of exchangeable NH4+ and pH in the soil solution, and their relation in the solution of Albaqualf soils with different salinity levels, under rice. Four field experiments were conducted with soils with exchangeable Na percentage (ESP) of 5.6, 9.0, 21.2, and 32.7 %. Prior to flooding, soil solution collectors were installed at depths of 5, 10 and 20 cm. The soil solution was collected weekly, from 7 to 91 days after flooding (DAF), to analyze exchangeable NH4+ and pH in the samples. Plant tissue was sampled 77 DAF, to determine N uptake and estimate the contribution of other N forms to rice nutrition. The content of exchangeable NH4+ decreased over time at all sites and depths, with a more pronounced reduction in soils with lower salinity levels, reaching values close to zero. A possible contribution of non-exchangeable NH4+ forms and N from soil organic matter to rice nutrition was observed. Soil pH decreased with time in soils with ESP 5.6 and 9.0 %, being positively correlated with the decreasing NH4+ levels at these sites.

Nutrition, dry matter accumulation and partitioning and phosphorus use efficiency of potato grown at different phosphorus levels in nutrient solution

High rates of phosphate fertilizers are applied to potato (Solanum tuberosum L.), which may cause antagonistic interactions with other nutrients and limit crop yields when over-supplied. The purpose of this study was to evaluate the influence of phosphorus (P) levels in nutrient solution on P use efficiency, nutritional status and dry matter (DM) accumulation and partitioning of potato plants cv. Ágata. The experiment was carried out in a greenhouse, arranged in a completely randomized block design with four replications. Treatments consisted of seven P levels in nutrient solution (0, 2, 4, 8, 16, 31, and 48 mg L-1). Plants were harvested after 28 days of growth in nutrient solution, and separated in roots, stems and leaves for evaluations. The treatment effects were analyzed by regression analysis. Phosphorus levels of up to 8 mg L-1 increased the root and shoot DM accumulation, but drastically decreased the root/shoot ratio of potato cv. Ágata. Higher P availability increased P concentration, accumulation and absorption efficiency, but decreased P use efficiency. Higher P levels increased the N, P, Mg, Fe, and Mn concentrations in roots considerably and decreased K, S, Cu, and Zn concentrations. In shoot biomass, N, P, K, and Ca concentrations were significantly increased by P applied in solution, unlike Mg and Cu concentrations. Although higher P levels (> 8 mg L-1) in nutrient solution increased P concentration, accumulation and absorption efficiency, the DM accumulation and partitioning of potato cv. Ágata were not affected.

Multiscale descriptions of density-driven flow instabilities in porous media

Les instabilités engendrées par des gradients de densité interviennent dans une variété d'écoulements. Un exemple est celui de la séquestration géologique du dioxyde de carbone en milieux poreux. Ce gaz est injecté à haute pression dans des aquifères salines et profondes. La différence de densité entre la saumure saturée en CO2 dissous et la saumure environnante induit des courants favorables qui le transportent vers les couches géologiques profondes. Les gradients de densité peuvent aussi être la cause du transport indésirable de matières toxiques, ce qui peut éventuellement conduire à la pollution des sols et des eaux. La gamme d'échelles intervenant dans ce type de phénomènes est très large. Elle s'étend de l'échelle poreuse où les phénomènes de croissance des instabilités s'opèrent, jusqu'à l'échelle des aquifères à laquelle interviennent les phénomènes à temps long. Une reproduction fiable de la physique par la simulation numérique demeure donc un défi en raison du caractère multi-échelles aussi bien au niveau spatial et temporel de ces phénomènes. Il requiert donc le développement d'algorithmes performants et l'utilisation d'outils de calculs modernes. En conjugaison avec les méthodes de résolution itératives, les méthodes multi-échelles permettent de résoudre les grands systèmes d'équations algébriques de manière efficace. Ces méthodes ont été introduites comme méthodes d'upscaling et de downscaling pour la simulation d'écoulements en milieux poreux afin de traiter de fortes hétérogénéités du champ de perméabilité. Le principe repose sur l'utilisation parallèle de deux maillages, le premier est choisi en fonction de la résolution du champ de perméabilité (grille fine), alors que le second (grille grossière) est utilisé pour approximer le problème fin à moindre coût. La qualité de la solution multi-échelles peut être améliorée de manière itérative pour empêcher des erreurs trop importantes si le champ de perméabilité est complexe. Les méthodes adaptatives qui restreignent les procédures de mise à jour aux régions à forts gradients permettent de limiter les coûts de calculs additionnels. Dans le cas d'instabilités induites par des gradients de densité, l'échelle des phénomènes varie au cours du temps. En conséquence, des méthodes multi-échelles adaptatives sont requises pour tenir compte de cette dynamique. L'objectif de cette thèse est de développer des algorithmes multi-échelles adaptatifs et efficaces pour la simulation des instabilités induites par des gradients de densité. Pour cela, nous nous basons sur la méthode des volumes finis multi-échelles (MsFV) qui offre l'avantage de résoudre les phénomènes de transport tout en conservant la masse de manière exacte. Dans la première partie, nous pouvons démontrer que les approximations de la méthode MsFV engendrent des phénomènes de digitation non-physiques dont la suppression requiert des opérations de correction itératives. Les coûts de calculs additionnels de ces opérations peuvent toutefois être compensés par des méthodes adaptatives. Nous proposons aussi l'utilisation de la méthode MsFV comme méthode de downscaling: la grille grossière étant utilisée dans les zones où l'écoulement est relativement homogène alors que la grille plus fine est utilisée pour résoudre les forts gradients. Dans la seconde partie, la méthode multi-échelle est étendue à un nombre arbitraire de niveaux. Nous prouvons que la méthode généralisée est performante pour la résolution de grands systèmes d'équations algébriques. Dans la dernière partie, nous focalisons notre étude sur les échelles qui déterminent l'évolution des instabilités engendrées par des gradients de densité. L'identification de la structure locale ainsi que globale de l'écoulement permet de procéder à un upscaling des instabilités à temps long alors que les structures à petite échelle sont conservées lors du déclenchement de l'instabilité. Les résultats présentés dans ce travail permettent d'étendre les connaissances des méthodes MsFV et offrent des formulations multi-échelles efficaces pour la simulation des instabilités engendrées par des gradients de densité. - Density-driven instabilities in porous media are of interest for a wide range of applications, for instance, for geological sequestration of CO2, during which CO2 is injected at high pressure into deep saline aquifers. Due to the density difference between the C02-saturated brine and the surrounding brine, a downward migration of CO2 into deeper regions, where the risk of leakage is reduced, takes place. Similarly, undesired spontaneous mobilization of potentially hazardous substances that might endanger groundwater quality can be triggered by density differences. Over the last years, these effects have been investigated with the help of numerical groundwater models. Major challenges in simulating density-driven instabilities arise from the different scales of interest involved, i.e., the scale at which instabilities are triggered and the aquifer scale over which long-term processes take place. An accurate numerical reproduction is possible, only if the finest scale is captured. For large aquifers, this leads to problems with a large number of unknowns. Advanced numerical methods are required to efficiently solve these problems with today's available computational resources. Beside efficient iterative solvers, multiscale methods are available to solve large numerical systems. Originally, multiscale methods have been developed as upscaling-downscaling techniques to resolve strong permeability contrasts. In this case, two static grids are used: one is chosen with respect to the resolution of the permeability field (fine grid); the other (coarse grid) is used to approximate the fine-scale problem at low computational costs. The quality of the multiscale solution can be iteratively improved to avoid large errors in case of complex permeability structures. Adaptive formulations, which restrict the iterative update to domains with large gradients, enable limiting the additional computational costs of the iterations. In case of density-driven instabilities, additional spatial scales appear which change with time. Flexible adaptive methods are required to account for these emerging dynamic scales. The objective of this work is to develop an adaptive multiscale formulation for the efficient and accurate simulation of density-driven instabilities. We consider the Multiscale Finite-Volume (MsFV) method, which is well suited for simulations including the solution of transport problems as it guarantees a conservative velocity field. In the first part of this thesis, we investigate the applicability of the standard MsFV method to density- driven flow problems. We demonstrate that approximations in MsFV may trigger unphysical fingers and iterative corrections are necessary. Adaptive formulations (e.g., limiting a refined solution to domains with large concentration gradients where fingers form) can be used to balance the extra costs. We also propose to use the MsFV method as downscaling technique: the coarse discretization is used in areas without significant change in the flow field whereas the problem is refined in the zones of interest. This enables accounting for the dynamic change in scales of density-driven instabilities. In the second part of the thesis the MsFV algorithm, which originally employs one coarse level, is extended to an arbitrary number of coarse levels. We prove that this keeps the MsFV method efficient for problems with a large number of unknowns. In the last part of this thesis, we focus on the scales that control the evolution of density fingers. The identification of local and global flow patterns allows a coarse description at late times while conserving fine-scale details during onset stage. Results presented in this work advance the understanding of the Multiscale Finite-Volume method and offer efficient dynamic multiscale formulations to simulate density-driven instabilities. - Les nappes phréatiques caractérisées par des structures poreuses et des fractures très perméables représentent un intérêt particulier pour les hydrogéologues et ingénieurs environnementaux. Dans ces milieux, une large variété d'écoulements peut être observée. Les plus communs sont le transport de contaminants par les eaux souterraines, le transport réactif ou l'écoulement simultané de plusieurs phases non miscibles, comme le pétrole et l'eau. L'échelle qui caractérise ces écoulements est définie par l'interaction de l'hétérogénéité géologique et des processus physiques. Un fluide au repos dans l'espace interstitiel d'un milieu poreux peut être déstabilisé par des gradients de densité. Ils peuvent être induits par des changements locaux de température ou par dissolution d'un composé chimique. Les instabilités engendrées par des gradients de densité revêtent un intérêt particulier puisque qu'elles peuvent éventuellement compromettre la qualité des eaux. Un exemple frappant est la salinisation de l'eau douce dans les nappes phréatiques par pénétration d'eau salée plus dense dans les régions profondes. Dans le cas des écoulements gouvernés par les gradients de densité, les échelles caractéristiques de l'écoulement s'étendent de l'échelle poreuse où les phénomènes de croissance des instabilités s'opèrent, jusqu'à l'échelle des aquifères sur laquelle interviennent les phénomènes à temps long. Etant donné que les investigations in-situ sont pratiquement impossibles, les modèles numériques sont utilisés pour prédire et évaluer les risques liés aux instabilités engendrées par les gradients de densité. Une description correcte de ces phénomènes repose sur la description de toutes les échelles de l'écoulement dont la gamme peut s'étendre sur huit à dix ordres de grandeur dans le cas de grands aquifères. Il en résulte des problèmes numériques de grande taille qui sont très couteux à résoudre. Des schémas numériques sophistiqués sont donc nécessaires pour effectuer des simulations précises d'instabilités hydro-dynamiques à grande échelle. Dans ce travail, nous présentons différentes méthodes numériques qui permettent de simuler efficacement et avec précision les instabilités dues aux gradients de densité. Ces nouvelles méthodes sont basées sur les volumes finis multi-échelles. L'idée est de projeter le problème original à une échelle plus grande où il est moins coûteux à résoudre puis de relever la solution grossière vers l'échelle de départ. Cette technique est particulièrement adaptée pour résoudre des problèmes où une large gamme d'échelle intervient et évolue de manière spatio-temporelle. Ceci permet de réduire les coûts de calculs en limitant la description détaillée du problème aux régions qui contiennent un front de concentration mobile. Les aboutissements sont illustrés par la simulation de phénomènes tels que l'intrusion d'eau salée ou la séquestration de dioxyde de carbone.

A Rotating black ring solution in five dimensions

The vacuum Einstein equations in five dimensions are shown to admit a solution describing a stationary asymptotically flat spacetime regular on and outside an event horizon of topology S1S2. It describes a rotating black ring. This is the first example of a stationary asymptotically flat vacuum solution with an event horizon of nonspherical topology. The existence of this solution implies that the uniqueness theorems valid in four dimensions do not have simple five-dimensional generalizations. It is suggested that increasing the spin of a spherical black hole beyond a critical value results in a transition to a black ring, which can have an arbitrarily large angular momentum for a given mass.

Exact solution to the mean exit time problem for free inertial processes driven by Gaussian white noise

We obtain the exact analytical expression, up to a quadrature, for the mean exit time, T(x,v), of a free inertial process driven by Gaussian white noise from a region (0,L) in space. We obtain a completely explicit expression for T(x,0) and discuss the dependence of T(x,v) as a function of the size L of the region. We develop a new method that may be used to solve other exit time problems.

Continued fraction solution for the radiative transfer equation in three dimensions

Starting from the radiative transfer equation, we obtain an analytical solution for both the free propagator along one of the axes and an arbitrary phase function in the Fourier-Laplace domain. We also find the effective absorption parameter, which turns out to be very different from the one provided by the diffusion approximation. We finally present an analytical approximation procedure and obtain a differential equation that accurately reproduces the transport process. We test our approximations by means of simulations that use the Henyey-Greenstein phase function with very satisfactory results.