A Fermi's problem is an order-of-magnitude problem (or order-of-magnitude estimate, order estimation), is an estimation problem designed to teach dimensional analysis or approximation (in this case approximation) of extreme scientific calculations, and such a problem is usually a back-of-the-envelope calculation (cit. Wikipedia)
Let's assume that a plant transpires 1 cm per day (just to exaggerate) per unit of area. Suppose this plant canopy covers an area of 100 m^2. The transpired volume in one day is ET = 0.01 * 100 = 1 m^3 (which is a lot, plants are reported to transpirate "hundred of liters", not cubic meters).
Now let's consider the specific hydraulic conductivity KS in Kg m^{-1} s^{-1} MPa^{-1}. According to Krober et al. (2014) and their database, the maximum hydraulic conductivity of Castanea Henryi is approximately (simplifying the numbers) 10 Kg m^{-1} s^{-1} MPa^{-1}. Skipping some details, the maximum sap flow, E_S, derived from this is of the same order of magnitude, expressed in Kg m^{-1} s^{-1} (hint: you need to calculate K(\psi) \psi, with K varying with psi, and psi being the pressure (in MPa) in the xylem, as in Manzoni et al., 2014).
To compare E_S and ET, I need to multiply E_S by the active trunk cross-sectional area CSA (according to Thurner) and divide it by the plant height (10 m) to account for the gradient. Then, I need to convert from Kg per second to Kg per day (multiplying by 10^5) and divide by the density of water to obtain the result in terms of volume (10^3 kg/m^3). Therefore:
E_S = 10 [ES value] * 10^5 [Seconds in a Day] * CSA [Cross-sectional Area] / 10^4 [Plant Height * Water Density] = 100 CSA
From ES = ET, it follows that:
CSA = 0.01 m^2
which could not be an unreasonable value (plant physiologists have to tell me). If the density measurement made by Kroeber et al. is actually related to the entire branch/trunk they used, it could mean that in a 1 m^2 stem (if the stem were 1 m^2), 1% contributes to the xylem flow. Unless I have forgotten any factor somewhere (which would be embarrassing, but I'll take the risk) or the measurements made by Kroeber et al. need to be adjusted differently.
According to Lüttschwager's study, this value would imply a much higher specific hydraulic conductivity than the KS observed in the outermost regions of the trunk where the flow is concentrated. Another consequence is that the less conductive species of this Chinese chestnut (38 out of 39 in the study) could only sustain such evaporation demands with much larger stems, which seems unreasonable, or a large percentage of vessels.
I would like to ask if the numbers I presented seem correct and reasonable to you, and if there is anything blatantly wrong in my reasoning or deduction from Kroeber's work (for those familiar with it) or elsewhere. Any comments are welcome.
P.S. - Most species in Kroeber's study have a KS that is 10 times smaller, which would require a CSA 10 times larger for the same evaporative demand.
References
Manzoni, Stefano, Giulia Vico, Gabriel Katul, Sari Palmroth, Robert B. Jackson, and Amilcare Porporato. 2013. “Hydraulic Limits on Maximum Plant Transpiration and the Emergence of the Safety-Efficiency Trade-Off.” The New Phytologist 198 (1): 169–78. https://doi.org/10.1111/nph.12126.
Kröber, Wenzel, Shouren Zhang, Merten Ehmig, and Helge Bruelheide. 2014. “Linking Xylem Hydraulic Conductivity and Vulnerability to the Leaf Economics Spectrum—A Cross-Species Study of 39 Evergreen and Deciduous Broadleaved Subtropical Tree Species.” PloS One 9 (11): e109211. https://doi.org/10.1371/journal.pone.0109211.
Lüttschwager, Dietmar, and Rainer Remus. 2007. “Radial Distribution of Sap Flux Density in Trunks of a Mature Beech Stand.” Annals of Forest Science 64 (4): 431–38. https://doi.org/10.1051/forest:2007020.
Thurner, Martin, Christian Beer, Thomas Crowther, Daniel Falster, Stefano Manzoni, Anatoly Prokushkin, and Ernst-Detlef Schulze. 2019. “Sapwood Biomass Carbon in Northern Boreal and Temperate Forests.” Global Ecology and Biogeography: A Journal of Macroecology 28 (5): 640–60. https://doi.org/10.1111/geb.12883.
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