Ohmic sap flow through trees

We're now in a position to model the entire flow of sucrose, from leaf to root. As we've established, the hydraulic flow across the xylem:phloem boundary, as well as down the phloem in the trunk, can be modeled as a given pressure giving rise to some flow, related by an effective resistivity $\mathcal{R}$ just as in electronic circuits.

In fact, we can take the analogy further.

The water loads the phloem from the xylem in the leaf. From here, it moves into the phloem of the trunk, and flows on to the sites of sucrose consumption, all the while riding the same osmotic pressure gradient.

This pressure across the vascular system is equivalent to a potential voltage kept across a circuit. Indeed, we can say that the xylem:phloem membrane, and the phloem are in series with one another, each opposing the flow of sap.

We can therefore calculate the fluid flow as if we have two resistors (one for the membrane and one for the stem) in series with a battery (the osmotic pressure):

$\Delta p = v_{sap}(\mathcal{R}_{mem} + \mathcal{R}_{stem})$

This idea is illustrated in the figure below:

As $\mathcal{R}_{mem}(l)$ is a function of leaf length and $\mathcal{R}_{stem}(h)$ is a function of tree height, we have an expression for the sap flow speed in terms of the size of the leaf $l$ and the height of the tree $h$:

$\begin{aligned} \displaystyle v_{sap} &= \frac{\Delta p}{(\mathcal{R}_{mem} + \mathcal{R}_{stem})} \\ &= \frac{\Delta p}{\frac{8h\eta}{\pi R^4} + \frac{1}{2\pi R l L_p}} \\ &= \frac{2 L_p R^4\pi l \Delta p}{R^3 + 16 h \eta l} \end{aligned}$

From here, we can take limiting cases and obtain simple expressions for interesting quantities such as the range possible of leaf sizes for a tree of a given height. Perhaps more interestingly, we can solve for the maximum possible height that a tree can have.

As the form indicates, the maximum possible sap flow rate is a saturating (i.e. asymptotic) function of leaf length, $l$. In theory, we can reach this point by having infinitely large leaves. However, building leaves itself costs energy, and so, at some point, we experience diminishing returns with increased leaf size.

leaf sizes

In the other direction, smaller leaves result in lower sucrose flow speeds and, at some point, the flow so generated fails to provide any advantage over theoretical plants of the same height without any vascular system, i.e. plants that depend solely on diffusion for sugar transport. It stands to reason that any circulatory system would have to at least outperform such theoretical plants to justify its usefulness.

Using this insight, we can calculate the range of possible leaf sizes as a function of a tree's height as well as the molecular details of the phloem and sap.

Next, let's calculate the maximum possible leaf size as a function of tree height.

Illustration by Maxicat Rhododendron