The fourth power

Water rises from roots to leaves through the xylem, a bundle of dead tubes pulled from above by evaporation. Sugars move from leaves to roots, fruits and growing tips through the phloem, living sieve tubes driven by pressure. Both are pipes, and the flow through a pipe is governed by the Hagen–Poiseuille law:

$$Q = \frac{\pi r^4\,\Delta P}{8\,\eta\,L},$$

where $Q$ is the volume flow, $r$ the radius, $\Delta P$ the pressure difference, $\eta$ the viscosity and $L$ the length. Because flow depends on the fourth power of the radius, a few wide vessels carry far more than many narrow ones, and a small narrowing has a large effect (Tyree and Zimmermann 2002).

import math

def poiseuille_R(r, length, visc):
    """Hydraulic resistance of a cylindrical tube, Pa·s/m³: R = 8ηL / (πr⁴)."""
    return 8 * visc * length / (math.pi * r ** 4)

# 1. Width matters to the fourth power: xylem vessels of different radius, 1 m long, water at 20 °C.
base = None
for r_um in (10, 20, 40, 80):
    g = 1 / poiseuille_R(r_um * 1e-6, 1.0, 1.0e-3)
    base = base or g
    print(f"vessel radius {r_um:2d} µm: conductance × {g / base:6.0f} relative to 10 µm")

# 2. Sap viscosity changes with temperature (water: 1.14 mPa·s at 15 °C, 0.80 at 30 °C).
print(f"\nwarm sap (30 °C) vs cool sap (15 °C): flow × {1.14 / 0.80:.2f} at the same pressure")

# 3. A sieve tube in the phloem: open lumen (r = 8 µm) interrupted every 250 µm by a sieve plate
#    1 µm thick with 60 pores. Phloem sap ≈ 1.7 mPa·s. Callose narrows the pores.
VISC, LUMEN_R, ELEMENT, PLATE, PORES = 1.7e-3, 8e-6, 250e-6, 1e-6, 60
def sieve_tube_R(pore_r, length=0.1):
    n = length / ELEMENT
    lumen = poiseuille_R(LUMEN_R, ELEMENT, VISC)
    plate = poiseuille_R(pore_r, PLATE, VISC) / PORES          # pores in parallel
    return n * (lumen + plate)

open_R = sieve_tube_R(1.0e-6)
for pore_um in (1.0, 0.75, 0.5, 0.25):
    R = sieve_tube_R(pore_um * 1e-6)
    print(f"sieve-pore radius {pore_um:4.2f} µm: flow at the same pressure = {100 * open_R / R:5.1f} % of open")
# vessel radius 10 µm: conductance ×      1 relative to 10 µm
# vessel radius 20 µm: conductance ×     16 relative to 10 µm
# vessel radius 40 µm: conductance ×    256 relative to 10 µm
# vessel radius 80 µm: conductance ×   4096 relative to 10 µm
#
# warm sap (30 °C) vs cool sap (15 °C): flow × 1.42 at the same pressure
# sieve-pore radius 1.00 µm: flow at the same pressure = 100.0 % of open
# sieve-pore radius 0.75 µm: flow at the same pressure =  68.3 % of open
# sieve-pore radius 0.50 µm: flow at the same pressure =  23.7 % of open
# sieve-pore radius 0.25 µm: flow at the same pressure =   1.8 % of open

Halving the radius of the sieve pores cuts the flow through the tube to about a quarter in this example. Narrowing them to a quarter of the radius nearly stops it. The law treats each pore as a long, smooth cylinder, which a sieve pore is not, so the numbers are approximate; the direction and the steepness are not.

Citrus greening

Huanglongbing, or citrus greening, is the most destructive disease of citrus in the world. It is associated with the bacterium Candidatus Liberibacter asiaticus, which lives in the phloem and is spread by a psyllid insect. Infected trees develop blotchy yellow leaves, lose roots and bear small, bitter fruit. There is no cure for infected trees in the field.

What carries over from Dendrobium to citrus

The proposal behind this chapter was that the same framework, open vessels, a controlled microclimate and heat treatment, could break the deadlock of citrus greening. Some of it does carry over. The physics of transport is the same in an orchid and a citrus tree. Leaf wetness, root aeration and salt stress matter in both. And heat can reduce the bacterium in young plants. But citrus greening is a vector-borne bacterial disease, and the measures with the strongest evidence remain control of the psyllid, healthy nursery stock and removal of infected trees. A microclimate framework could support those measures, for example in screened nurseries where heat treatment of young plants is practical, but it does not replace them, and it should not be presented to citrus growers as a solution.