Experimental and computational comparison of freeze-thaw induced pressure generation in red and sugar maple

TL;DR

An existing model that describes heat and mass transport within the multiphase gas–liquid–ice mixture in the porous xylem tissue reproduces realistic exudation behavior, thereby providing novel insights into the specific physical mechanisms that dominate positive pressure generation in maple trees.

Abstract

Sap exudation is the process whereby trees such as sugar (Acer saccharum) and red maple (Acer rubrum) generate unusually high positive stem pressure in response to repeated cycles of freeze and thaw. This elevated xylem pressure permits the sap to be harvested over a period of several weeks and hence is a major factor in the viability of the maple syrup industry. The extensive literature on sap exudation documents competing hypotheses regarding the physical and biological mechanisms that drive positive pressure generation in maple, but to date relatively little effort has been expended on devising mathematical models for the exudation process. In this paper, we utilize an existing model of Graf et al. [J. Roy. Soc. Interface 12:20150665, 2015] that describes heat and mass transport within the multiphase gas-liquid-ice mixture in the porous xylem tissue. The model captures the inherent multiscale nature of xylem transport by including phase change and osmotic transport in wood cells on the microscale, which is coupled to heat transport through the tree stem on the macroscale. A parametric study based on simulations with synthetic temperature data identifies the model parameters that have greatest impact on stem pressure build-up. Measured daily temperature fluctuations are then used as model inputs and the resulting simulated pressures are compared directly with experimental measurements taken from mature red and sugar maple stems during the sap harvest season. The results demonstrate that our multiscale freeze-thaw model reproduces realistic exudation behavior, thereby providing novel insights into the specific physical mechanisms that dominate positive pressure generation in maple trees.

Figures (7)

• Figure 1: Photograph of a taphole containing a black nylon spout that is connected by plastic tubing to an Omega PX-26-030GV pressure sensor. Additional wires leading to thermocouples are also shown.
• Figure 2: Sapwood microstructure and the idealized 2D model geometry. (a) A microscopic cut-away view of the sapwood within a typical hardwood tree, depicting the vessels and (libriform) fibers that are central to the model. Tracheids are connected hydraulically to neighboring vessels via paired pits, which is why they are "lumped together" in our model with vessels. Note that fiber walls also contain pits, but they are unpaired and hence unconnected to adjacent vessels or tracheids. (b) A single fiber--vessel pair showing the main geometric parameters. The horizontal cutting plane highlights the planar cross-section corresponding to the 2D model geometry in Figure \ref{['fig:cell-geometry']}c. The dashed circles represent the $N$ copies of the fiber that are incorporated into the equations through a simple multiplier $N$. (c) The 2D model geometry depicting a thawing scenario. A thawing fiber of radius $R^f$ (containing nested layers of gas, ice and liquid water) is located adjacent to a thawed vessel of radius $R^v$ (containing gas and liquid sap). As the fiber ice layer thaws, the fiber gas bubble expands and forces melt-water through the porous wall into the vessel at a rate $U$, thereby compressing the vessel gas and increasing the vessel sap pressure.
• Figure 3: The freeze--thaw process within a circular tree stem cycles between four main stages (i $\rightarrow$ ii $\rightarrow$ iii $\rightarrow$ iv $\rightarrow$ i $\rightarrow$ …) as ambient temperature $T$ cycles below and above the freezing point: (i) completely thawed (with $T>0$); (ii) partially frozen ($T\searrow 0$), with a freezing front advancing radially inward to the center of the stem; (iii) completely frozen ($T<0$); (iv) partially thawed ($T\nearrow 0$), with a thawing front advancing radially inward. The freezing/thawing fronts in (ii,iv) are thin annular regions (shaded in grey, and in reality much thinner than depicted here) wherein the liquid is in a "mixed" state; that is, the water in the fibers is freezing/frozen and the sap in the vessels thawing/thawed. The thawing front circled on the left of (iv) is magnified in Figure \ref{['fig:cell-geometry']}c to the cellular scale, which depicts an individual vessel and an adjacent fiber in a partially thawed state.
• Figure 4: (a) The reference cell $\mathcal{Y}$ containing a fiber located at the center (the brown dashed line is the fiber wall) outside of which lies the vessel. For the purposes of the periodic homogenization process, an artificial boundary $\Gamma$ (outer dotted circle) is introduced that separates $\mathcal{Y}$ into two sub-regions: $\mathcal{Y}^2$, a fiber--vessel overlap region where heat diffusion is slow (shaded in medium blue); and $\mathcal{Y}^1$, the outer portion of the vessel region where diffusion is relatively fast (light blue). A gas bubble of radius $r$ is depicted in the lower-left corner, and the root water source $U_r$ in the lower-right. (b) An annular sapwood cross-section is tiled periodically with copies of the reference cell from (a). Mature trees contain a non-conducting heartwood region extending out to radius $x=R_{\text{\rmfamilysap}}$, whereas younger saplings may have no heartwood ($R_{\text{\rmfamilysap}}=0$).
• Figure 5: (Left, a-i and b-i) Measured air temperatures are plotted for two red maple trees (R1, R2) and one sugar maple (S1) from the UVM experiments. The temperature plot for sugar maple in (b-i) also includes values of soil temperature (at 30 cm depth) which remain mostly above 0${}^\circ$C, hence supporting the assumption that liquid water is present even when air temperatures are below freezing. The raw temperature data (blue points) are regularized by applying a simple weighted-average smoothing -- the resulting smoothed data are shown alongside the original temperatures in the zoomed views (a-i,zoom, b-i,zoom). (Right, a-ii, a-iii and b-ii) Corresponding pressure data for the three trees. An extra set of pressure measurements is included in the sugar maple plot (b-ii) to illustrate the impact of taking measurements on the north/south sides of the stem.