Estimating volume flow rates through xylem conduits

We discuss the errors in common approximations of the volume flow rate for laminar flow through conduits with noncircular transverse sections. Before calculating flow rates, ideal geometric shapes are chosen to represent the noncircular transverse sections. The Hagen-Poiseuille equation used with hydraulic diameter underestimates the volume flow rate for laminar flow through conduits even with such ideal shapes. Correction factors that have been proposed for the Hagen-Poiseuille equation also lead to underestimates of the volume flow rate for those shapes. The exact solutions are sometimes difficult to attain, but rates calculated using the exact solutions for the ideal shapes may be as much as five times higher than the approximated rates for common transversely elongated shapes. Either the exact solutions or the approximations may be used to calculate volume flow rates through the xylem of plants. Both of these methods actually approximate flow through the original conduits because the shapes used are approximations of the conduits' transverse sections. We recommend using the exact solutions whenever possible; they should be closer to the real solution than other approximations. We give tables of correction factors for use in the cases where calculating volume flow rate from the approximate solution, the Hagen-Poiseuille equation, is more feasible. Obtaining theoretical volume flow rates that are larger than previously thought highlights the need to clarify the causes of differences between the theoretical rates and the smaller measured volume flow rates in plant xylem. As part of an ongoing project to understand the dynamics of water flow through the xylem we discuss exact equations to calculate water flow through noncircular ideal capillaries and investigate the errors in frequently used approximations. In determining theoretical volume flow rates through the xylem of plants, the Hagen-Poiseuille equation for laminar flow through conduits is often used. The Hagen-Poiseuille equation provides the exact solution for laminar flow in individual ideal capillaries with circular transverse sections, but the application of this equation to conduits with noncircular transverse sections using the hydraulic diameter results in errors in the calculated volume flow rates. Exact solutions for the determination of volume flow rate through capillaries with some common noncircular transverse sections have recently been published in English (White, 1991). In general, errors in calculated flow rates may result from anatomical deviations of the xylem conduits from ideal capillaries, or from methods used to calculate the volume flow rate. This paper investigates the latter more completely than previous botanical literature (e.g., Leyton, 1975; Nonweiler, 1975; Petty, 1978; Pickard, 1981; Zimmermann, 1983; Calkin, Gibson, and Nobel, 1986; Lewis, 1992), discusses some exact solutions for volume flow rates through some common noncircular conduits, .and recommends methods for the calculation of theoretical volume flow rates based on desired accuracy, speed, and ease of calculation.