Modeling Fluid Dynamics in Xylem Vessels

The hexagonal xylem vessel model (Fig 7) can analyze the flow characteristics by the energy conservation law (Bernoulli equation). The flow between arbitrary sections satisfies the Bernoulli equation, which was written in sections from the inlet to the exit sections Z₁, Z₂, ···, Z_n as:
$$\begin{array}{c}\frac{P_1}{\rho g}+\frac{V_1^2}{2g}+z_1=\frac{P_2}{\rho g}+\frac{V_2^2}{2g}+z_2+{\xi }_1\frac{V_2^2}{2g}+\lambda \frac{l_1{V}_2^2}{8Dg}\\ \frac{P_2}{\rho g}+\frac{V_2^2}{2g}+z_1=\frac{P_3}{\rho g}+\frac{V_3^2}{2g}+z_3+{\xi }_2\frac{v_3^2}{2g}+\lambda \frac{l_2{V}_3^2}{8Dg}\\ \cdot \cdot \cdot \cdot \cdot \cdot \\ \cdot \cdot \cdot \cdot \cdot \cdot \\ \frac{P_{n-1}}{\rho g}+\frac{V_{n-1}^2}{2g}+z_{n-1}=\frac{P_n}{\rho g}+\frac{V_n^2}{2g}+z_n+{\xi }_{n-1}\frac{V_n^2}{2g}+\lambda \frac{l_{n-1}{V}_n^2}{8Dg}\end{array}$$
Where P_n and V_n were the average pressure and flow velocity at section n, ρ was fluid density, g was the acceleration of gravity, Z_n was the position head of water at the section, ξ_n−1 was the local loss coefficient of section n-1 to section n, λ was friction factor of head loss, l_n−1 was the length between two adjacent sections. D was the hydraulic radius of the xylem vessel, the expression of D was:
$$D=\frac{A}{\chi }$$
Add the two sides of the equations of Eq (1) in order:
$$\frac{P_1-P_n}{\rho g}=z_n-z_1+{\xi }_1\frac{V_2^2}{2g}+{\xi }_2\frac{V_3^2}{2g}+\cdots +{\xi }_{n-1}\frac{V_n^2}{2g}+\lambda \frac{L{V}_n^2}{8Dg}$$
Where l₁+l₂+l₃+···+l_n-1= L, L was the total length of the xylem vessel.
Known by the continuity equation:
$$V_1{A}_1=V_2{A}_2=V_3{A}_3=\cdots =V_n{A}_n$$
In Eq (4), A_i(i = 1,2…,n) was the flow area at the corresponding section, Substituting Eq (4) into Eq (3) give:
$$\frac{\Delta P}{\rho g}=L+\left[\lambda {\left(\frac{A_1}{A_n}\right)}^2\frac{L}{4D}+{\displaystyle \sum _{i=1}^{n-1}{\xi }_i{\left(\frac{A_1}{A_{i+1}}\right)}^2}\right]\frac{V_1{}^2}{2g}$$
Where
$$\xi =\left[\lambda {\left(\frac{A_1}{A_n}\right)}^2\frac{L}{4D}+{\displaystyle \sum _{i=1}^{n-1}{\left(\frac{A_1}{A_{i+1}}\right)}^2}{\xi }_i\right]$$
Eq (6) was simplified to:
$$\frac{\Delta P}{\rho g}=L+\xi \frac{V_1^2}{2g}$$
Expressed as:
$$\xi =\frac{2}{V_1^2}\left(\frac{\Delta P}{\rho }‐Lg\right)$$
Expressed by flow rate:
$$\xi =\frac{24R^4}{q^2}(\frac{\Delta P}{\rho }-Lg)$$
In Eqs (8A, 8B), was the flow resistance coefficient of hexagonal xylem vessel, q was the average flow rate.
For pentagon, quadrilateral and circular xylem vessel model, the expressions were:
$$\xi =\frac{50R^4{(\tan {36}^{\circ })}^2}{q^2}(\frac{\Delta P}{\rho }-Lg)$$
$$\xi =\frac{32R^4}{q^2}(\frac{\Delta P}{\rho }-Lg)$$
$$\xi =\frac{2{\pi }^2{R}^4}{q^2}(\frac{\Delta P}{\rho }-Lg)$$