Geoscience Reference
In-Depth Information
The low velocity
v
(m s
-1
) in capillary tubes such as xylem vessels can be calcu-
lated with Poiseuille's law (Koorevaar et al.,
1983
):
∂
∂
H
x
2
r
p
v
=−
(6.19)
8
η
where
r
(m) is the radius of the tube,
η
is the dynamic viscosity (≈ 0.001 Pa s),
H
p
is
the pressure equivalent of the hydraulic potential (Pa) and
x
is the low direction. For
example, take a sap low velocity
v
of 10.8 m h
-1
= 0.003 m s
-1
. When
r
= 50
μ
m as
in many trees, Eq. (
6.19
) yields 10 kPa m
-1
= 0.1 bar m
-1
for the equivalent pressure
gradient. This pressure gradient is valid for an ideal tube with smooth walls. With the
plant vessels one has to expect a higher low resistance, caused by the roughness of
vessel walls and presence of perforation plates. Therefore the pressure gradient has to
be greater to attain
v
= 10.8 m h
-1
, approximately 1.1 × 0.1 = 0.11 bar m
-1
. Note that
this gradient would allow water to low through a horizontal pipe (Ehlers and Goss,
2003
).
With plants growing upright, the gravitational head also has to be considered when
calculating the gradient necessary for the suggested water lux in the vertical direc-
tion. The gradient for the gravitational head is 10 kPa m
-1
= 0.1 bar m
-1
of plant
height. Therefore the total hydraulic head gradient in vertical trees is ca. 0.11 + 0.10 =
0.21 bar m
-1
. For a 30 m high tree we calculate a hydraulic head difference of 6.3 bar
between the soil surface and the canopy, and for a coastal sequoia from California of
100 m height or a karri tree of the same size from the south coast of Western Australia
a difference of at least 21.0 bar. How can plants generate these giant hydraulic head
differences?
Question 6.4:
Calculate the hydraulic head loss (m) due to low friction and due to
gravity in case of a 20 m high tree. The xylem vessels have a radius
r
= 50
μ
m. The sap
low velocity
v
= 0.002 m s
-1
. Increase the friction head loss according to Eq. (
6.19
) with
a factor of 1.1 due to the roughness of vessel walls and the perforation plates.
For quite a long time it was thought that the hydraulic head difference in the vessels
was caused by excess pressure in the roots. Today we know that the hydraulic head
differences of up to 210 m, as calculated for the sequoia, are not caused by pressure
but by pull. That is pull by a negative hydraulic head in the canopy. Meteorologi-
cal factors control the relative air humidity in the intercellular spaces near the sto-
mata. For instance at a relative air humidity of 98%, the hydraulic head will be as
low as -135.7 m. The decline in hydraulic head within the soil-plant-atmosphere
continuum is drawn in
Figure 6.10
for four hypothetical situations. Case 1 depicts
the hydraulic head decline when the soil is moist. Within the mesophyll cells of the
leaf (DE), the hydraulic head stays much above the critical limit of about -200 m,
below which the plant will start wilting. In case 2 the transpiration rate is greater at
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