A two-scale Stefan problem arising in a model for tree sap exudation

The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze-thaw cycles, gives rise to an interesting multi-scale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the global temperature on the scale of the tree stem, with all the remaining physics relegated to equations defined on the reference cell. We derive a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. Numerical simulations are then performed to validate the results and draw conclusions regarding the phenomenon of sap exudation, which is of great importance in trees such as sugar maple and a few other related species. The particular form of our homogenized temperature equation is obtained using periodic homogenization techniques with two-scale convergence, which we investigate theoretically in the context of a simpler two-phase Stefan-type problem corresponding to a periodic array of melting cylindrical ice bars with a constant thermal diffusion coefficient. For this reduced model, we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, clearly identifying the missing pieces required to extend the proofs to the fully nonlinear sap exudation model. Numerical simulations of the reduced equations are then compared with results from the complete sap exudation model.Comment: 35 pages, 8 figures. arXiv admin note: text overlap with arXiv:1411.303

A three-phase free boundary problem with melting ice and dissolving gas

We develop a mathematical model for a three-phase free boundary problem in one dimension that involves the interactions between gas, water and ice. The dynamics are driven by melting of the ice layer, while the pressurized gas also dissolves within the meltwater. The model incorporates a Stefan condition at the water-ice interface along with Henry's law for dissolution of gas at the gas-water interface. We employ a quasi-steady approximation for the phase temperatures and then derive a series solution for the interface positions. A non-standard feature of the model is an integral free boundary condition that arises from mass conservation owing to changes in gas density at the gas-water interface, which makes the problem non-self-adjoint. We derive a two-scale asymptotic series solution for the dissolved gas concentration, which because of the non-self-adjointness gives rise to a Fourier series expansion in eigenfunctions that do not satisfy the usual orthogonality conditions. Numerical simulations of the original governing equations are used to validate the series approximations

TAMM review : On the importance of tap and tree characteristics in maple sugaring

Maple sugaring mainly uses sugar and red maples (Acer saccharum and Acer rubrum) by tapping them for sap in the leafless-state across large portions of their ranges. How much sap exudes from a tap hole and how sweet this sap is, can vary substantially. Year-to-year variation in sap yield and sugar content can be primarily traced to differences in meteorological conditions that drive sap runs. Yet, how much of the total variation in sap yield and sugar content is linked to the year, site, species, tree, or tap has not been investigated systematically. Here, we reviewed the literature and also compiled a dataset of sap yield and sugar content from gravity taps on 324 red and sugar maples. The compiled data originates from multiple studies at ten sites across a large proportion of the ranges of sugar and red maple and stretches over eleven years. Using about 15 000 data points on sap yield and sap sugar content, we analysed the importance of tap and tree characteristics, such as height of the tap hole on the stem or diameter at breast height. We also review previous research on the importance of tap and tree characteristics in maple sugaring. Moreover, we partition variability in the data to attribute it to species, site, tree, year, and tap characteristics. Our results indicate that species, site and tree characteristics are the three largest sources of variability with regards to sap yield and the sap’s sucrose concentration. However, differences between years and tap characteristics, which were found to be comparatively minor sources of variability in sap yield and the sap’s sucrose concentration, have attracted far more attention in the past. We advocate for the continuation and expansion of systematic measurements of sap characteristics across a network of sites to further improve our understanding of maple sugaring. Such an understanding will be instrumental to prepare maple sugaring operations against the imminent effects of the climate and biodiversity crises and ensure their sustainability to perpetuate this traditional activity

Étude de l'impact des classes de sirop d'érable sur leurs propriétés physicochimiques pendant l'entreposage et leur comportement à l'ébullition et au refroidissement

Les produits de l'érable sont fabriqués généralement par ébullition, refroidissement, puis agitation du sirop d'érable « Canada catégorie A ». Les cristaux de sucrose formés donnent la texture au produit. En confiserie, la courbe d'élévation du point d'ébullition de l'eau (EPE) en fonction des solides solubles permet d'arrêter l'ébullition au niveau approprié de concentration pour le produit visé, alors que la courbe de la température de transition vitreuse selon la teneur en solides solubles permet de recommander des paramètres de refroidissement. Les outils actuels se basent principalement sur des solutions aqueuses de sucrose, mais la présence d'autres composés les influencent. Le sirop d'érable est une solution aqueuse, surtout de sucrose, mais aussi de sucre inverti, de polyphénols et de composés azotés. Leur teneur varie selon la couleur du sirop d'érable, représentée par sa classe : « Doré », « Ambré », « Foncé » ou « Très foncé ». Les caractéristiques physicochimiques pourraient changer au cours de leur entreposage. Au cours de ce projet, les effets de la durée d'entreposage sur une année et de la classe de sirop sur leurs caractéristiques physicochimiques ont été étudiés. Les teneurs en polyphénols, sucre inverti et composés azotés tendaient à être plus élevées dans les sirops plus foncés, contrairement à la concentration en sucrose qui diminuait légèrement. Au cours de l'entreposage, la teneur en sucrose augmentait, alors que les concentrations en polyphénols et le pH diminuaient. Pour mieux comprendre l'effet des classes de sirop pendant leur transformation, leur comportement a été étudié pendant l'ébullition et le refroidissement. La courbe de la température de transition vitreuse était commune à tous les sirops, tandis que celles de l'EPE variaient selon leur classe. Ces connaissances appuieront une meilleure compréhension de la cristallisation du sucrose dans un contexte de transformation du sirop d'érable.Maple products are generally made from "Canada grade A" maple syrup by boiling, cooling, and then stirring. Crystallization gives the characteristic texture to maple products. In confectionery, the Boiling point elevation (BPE) depending on total soluble solids curve is used to stop boiling at the appropriate concentration level for the intended product, while the curve of glass transition temperature depending on total soluble solids is used to recommend cooling parameters. Current tools are mainly based on aqueous solutions of sucrose, but the addition of other compounds influences them. Maple syrup is an aqueous solution, mainly composed of sucrose, but also of invert sugar, total phenolic content and nitrogenous compounds. Their content changes according to the color of the maple syrup, represented by its class: " Golden ", " Amber ", " Dark " or " Very dark ". Their physicochemical characteristics could change during storage. During this project, the effects of storage time over a year and syrup class on their physicochemical characteristics were studied. Polyphenols, invert sugar and nitrogen compounds tended to be higher in darker syrups, while the sucrose concentration decreased slightly. During storage, sucrose content increased, while polyphenol concentrations and pH decreased. To better understand the effect of the syrup classes during processing, their behavior during boiling and cooling was studied. The glass transition temperature curve was common to all syrups, while the BPE curves varied according to their class. This knowledge will be used to better understand sucrose crystallization in a maple syrup processing context