We develop a mathematical model for sap exudation in a maple tree that is based on a purely physical mechanism for internal pressure generation in trees in the leafless state. There has been a long-standing controversy in the tree physiology literature over precisely what mechanism drives sap exudation, and we aim to cast light on this issue. Our model is based on the work of Milburn and O'Malley [Can. J. Bot., 62(10): [2101][2102][2103][2104][2105][2106] 1984] who hypothesized that elevated sap pressures derive from compressed gas that is trapped within certain wood cells and subsequently released when frozen sap thaws in the spring. We also incorporate the extension of Tyree [in Tree Sap, pp. 37-45, eds. M. Terazawa et al., Hokkaido Univ. Press, 1995] who argued that gas bubbles are prevented from dissolving because of osmotic pressure that derives from differences in sap sugar concentrations and the selective permeability of cell walls. We derive a system of differential-algebraic equations based on conservation principles that is used to test the validity of the Milburn-O'Malley hypothesis and also to determine the extent to which osmosis is required. This work represents the first attempt to derive a detailed mathematical model of sap exudation at the micro-scale.

Sap transport in trees has long fascinated scientists, and a vast literature exists on experimental and modelling studies of trees during the growing season when large negative stem pressures are generated by transpiration from leaves. Much less attention has been paid to winter months when trees are largely dormant but nonetheless continue to exhibit interesting flow behaviour. A prime example is sap exudation, which refers to the peculiar ability of sugar maple (Acer saccharum) and related species to generate positive stem pressure while in a leafless state. Experiments demonstrate that ambient temperatures must oscillate about the freezing point before significantly heightened stem pressures are observed, but the precise causes of exudation remain unresolved. The prevailing hypothesis attributes exudation to a physical process combining freeze-thaw and osmosis, which has some support from experimental studies but remains a subject of active debate. We address this knowledge gap by developing the first mathematical model for exudation, while also introducing several essential modifications to this hypothesis. We derive a multiscale model consisting of a nonlinear system of differential equations governing phase change and transport within wood cells, coupled to a suitably homogenized equation for temperature on the macroscale. Numerical simulations yield stem pressures that are consistent with experiments and provide convincing evidence that a purely physical mechanism is capable of capturing exudation.

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In this paper, we propose a new mathematical model describing the effect of phosphocitrate (PC) on sodium sulphate crystallization inside bricks. This model describes salt and water transport, and crystal formation in a one dimensional symmetry. This is a preliminary study that takes into account mathematically the effects of inhibitors inside a porous stone. To this aim, we introduce two model parameters: the crystallization rate coefficient, which depends on the nucleation rate, and the specific volume of precipitated salt. These two parameters are determined by numerical fitting of our model for both the case of the brick treated with PC and non treated one

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