A two-dimensional heat transfer model for predicting freeze-thaw events in sugar maple trees

Highlights

• Numerical model is consistent with experimentally measured tree temperatures.

• Frequency of freeze-thaw cycles increases with decreasing stem diameter.

• Plantation method a promising strategy for maple syrup production in mild climates.

Abstract

Freeze-thaw cycles, where temperatures fluctuate above and below 0 °C, are the cause of elevated stem pressures that drive sap flow from within the sugar maple. This temperature-dependency has historically limited the production of maple syrup to select regions of North America. The plantation method of sap harvesting (which uses densely planted saplings instead of mature trees) now raises the possibility of a New Zealand-based maple syrup industry. In this study, a transient 2D heat transfer model was developed to predict freeze-thaw events in trees, and thereby evaluate potential plantation locations based on their climate. The heat transfer phenomena which have been modelled are bulk thermal diffusion, diffusion across discrete wood layers, convection, infrared radiation and solar radiation. Through experimental validation the model was found capable of predicting temperatures in real-life trees with high accuracy. Sensitive parameters were the bark absorptivity and sapwood diffusivity. Simulation results also indicate that the frequency of freeze-thaw cycles increase dramatically in saplings, as compared to mature trees, making maple syrup production potentially viable in locations that would otherwise fail when using traditional methods.

Introduction

In the springtime, freezing and thawing of the sugar maple (Acer saccharum) gives rise to elevated stem pressures that, upon wounding of the tree, will cause its sap to flow readily. For producers of maple syrup, harvesting the sap is a simple matter of drilling a hole into the wood, inserting a tap and directing the exuded sap into a container, provided that the ambient temperature fluctuates above and below 0°C over a period of several consecutive days. This process is known as a ‘freeze-thaw cycle’. The sap is then boiled and concentrated into maple syrup. Currently, the world's only commercial producers of maple syrup are found in the north-eastern United States and south-eastern Canada, wherein the sugar maple grows abundantly and the climate is conducive to sap exudation. In fact, 75% of the global maple syrup supply comes from Quebec alone (Houle et al., 2015). Researchers in New Zealand are now interested in creating a domestic maple syrup industry, motivated by newly-discovered methods to harvest sap from the excised stems of sugar maple saplings (as opposed to mature trees) under vacuum (Perkins and van den Berg, 2019). The underlying assumption is that smaller diameter saplings will allow for reliable freeze-thaw cycles to be established in New Zealand's milder climate. Not only would this venture diversify the maple syrup industry, which is currently susceptible to pests, disease and extreme weather events, as well as climate change (Houle et al., 2015), but it is also of economic interest (Driller et al., 2018). The knowledge that freeze-thaw cycles are needed to induce sap flow in sugar maple lends itself towards a predictive heat transfer model, through which the importance of various meteorological factors can be assessed.

The mechanism responsible for maple sap exudation has long been a source of debate. It is thought to be the combined effects of 1) a maple tree's unique cellular structure, 2) the presence of dissolved sucrose in the sap and 3) ice formation during a freeze-thaw cycle. The leading theory was first proposed by Milburn and O'Malley (1984) and later expanded upon by Tyree (1995). Various mathematical models have since been developed in an attempt to validate their work (Ceseri and Stockie, 2013, Graf and Stockie, 2014, Graf et al., 2015). Recently, Graf et al. (2015) were successful in replicating the pressure build-up in a black walnut tree (a closely-related species) when subjected to a freeze-thaw cycle. This was achieved by coupling a model of a sugar maple's unique cellular structure with a 1-D tree heat transfer model using a technique known as periodic homogenisation. In the heat transfer model a number of simplifying assumptions were made. For example, radial symmetry was imposed on the tree, and the bark was assumed to be in thermal equilibrium with the surrounding air. However, conventional wisdom says that in the northern hemisphere, south facing tap-holes usually flow with greater volumes of sap due to their heightened sun exposure (Wiegand, 1906). Clearly the pattern of sap flow is radially asymmetric and related to the short-term temperature history of the tree, including solar radiation. It appears that all models specific to maple sap exudation have failed to account for solar effects thus far.

Other limited attempts have been made to model heat transfer within trees. In 1966 Derby and Gates (1966) analysed temperatures in a two-dimensional cross section of an aspen tree using the finite-difference method. The authors recognised both the heterogeneous and anisotropic nature of wood; bark, sapwood and heartwood each have different thermal properties, and conductivities in the axial direction are typically 2–2.5 times greater than in the radial or tangential directions (Herrington, 1969). External processes such as solar radiation, infrared radiation and convection were also modelled, but their analysis was limited by the computing power available at the time. Costa et al. (1991) developed a model to predict stem temperatures under fire conditions, utilising the finite-volume method and a similar 2D geometry. Chatziefstratiou et al. (2013) added further complexity by simulating the change in wood's properties when exposed to high temperatures. Outside the realm of forest fires, however, tree heat transfer models have seen little advancement. The work of Potter and Andresen (2002) remains the most relevant to date. Their model was able to replicate some qualitative features, like the timing of peak temperatures in an aspen and pine tree, but the prediction itself deviated by up to 8 °C. Furthermore, by their own admission the implementation of solar radiation was relatively crude.

In this paper we present an updated finite-difference model of heat transfer within a tree. We place emphasis on simulating the natural processes that affect a tree's diurnal temperature cycle, including that of incident solar radiation, infrared radiative exchange and convection. A solar angles calculator is used to apply appropriate boundary conditions given the time of day, aspect angle and measured sunlight intensity. The model is also designed to take minimal inputs and be compatible with meteorological data from weather stations, so that the climate in locations of interest may be simulated with relative ease. We then demonstrate a unique application; predicting the occurrence of freeze-thaw events in sugar maples. In doing so, we hope to not only provide a useful tool for optimising sap yields, but also contribute to the lack of research on heat transfer in trees in general.