We can imagine the xylem formed by ducts of 0.02 mm radius that in their upper part are closed by the leaves, which have pores of 5 nm radius. Determine: (a) the flow rate through a conduit in a tree 30 m high, if the pressure at the top is ten percent greater than that necessary to maintain the water column, (b) the pressure necessary to maintain the same flow rate in a set of vessels with a radius of 5 nm and whose total area was equal to that of the xylem vessel.
Question
We can imagine the xylem formed by ducts of 0.02 mm radius that in their upper part are closed by the leaves, which have pores of 5 nm radius. Determine: (a) the flow rate through a conduit in a tree 30 m high, if the pressure at the top is ten percent greater than that necessary to maintain the water column, (b) the pressure necessary to maintain the same flow rate in a set of vessels with a radius of 5 nm and whose total area was equal to that of the xylem vessel.
Solution
To solve this problem, we need to use the Poiseuille's law, which states that the flow rate (Q) through a cylindrical pipe is directly proportional to the fourth power of the radius (r), the pressure difference (ΔP), and inversely proportional to the length (L) and the viscosity (η) of the fluid. The equation is:
Q = (πr⁴ΔP) / (8ηL)
(a) To find the flow rate through a conduit in a tree 30 m high, we need to know the pressure difference (ΔP). The problem states that the pressure at the top is ten percent greater than that necessary to maintain the water column. The pressure necessary to maintain a water column of height h is given by ρgh, where ρ is the density of water (1000 kg/m³), g is the acceleration due to gravity (9.81 m/s²), and h is the height of the water column (30 m). So, ΔP = 1.1ρgh - ρgh = 0.1ρgh = 0.1 * 1000 kg/m³ * 9.81 m/s² * 30 m = 29430 Pa. The viscosity of water (η) is approximately 1.002 x 10^-3 Pa.s. Substituting these values into the Poiseuille's law, we get:
Q = (π * (0.02 m)⁴ * 29430 Pa) / (8 * 1.002 x 10^-3 Pa.s * 30 m) = 0.000000145 m³/s
(b) To find the pressure necessary to maintain the same flow rate in a set of vessels with a radius of 5 nm, we need to rearrange the Poiseuille's law to solve for ΔP:
ΔP = (8ηLQ) / (πr⁴)
The total area of the set of vessels is equal to that of the xylem vessel, so the number of vessels is (0.02 m / 5 x 10^-9 m)² = 1.6 x 10^16. Since the flow rate is divided equally among all vessels, the flow rate through each vessel is 0.000000145 m³/s / 1.6 x 10^16 = 9.0625 x 10^-24 m³/s. Substituting these values into the rearranged Poiseuille's law, we get:
ΔP = (8 * 1.002 x 10^-3 Pa.s * 30 m * 9.0625 x 10^-24 m³/s) / (π * (5 x 10^-9 m)⁴) = 1.38 x 10^9 Pa.
Similar Questions
Which of the following describes how water moves through the xylem of a plant?A.In a downward direction through pressure-flow.B.In an upward direction through active transport.C.In a downward direction through capillary action.D.In an upward direction through the force of adhesion.
Which factors contribute to water movement through xylem?A.Adhesion and pressure-flow hypothesisB.Transpiration and capillary actionC.Transpiration and pressure-flow hypothesisD.capillary action and diffusion
What is the Pressure Flow Hypothesis? *1 pointTransport of xylem sap through xylem from the source cell to the sink cell.Transport of xylem sap through xylem from the sink cell to the source cell.Transport of phloem sap through sieve tube from the source cell to the sink cell.Transport of phloem sap through sieve tube from sink cell to source cell.
Which of the following factors contributes to capillarity in xylem vessels?Repulsion between water and xylem cell wallsAdhesive force between water and xylem cell wallsAttraction between water and airTranspiration pull
In xylem the function of conducting water and providing mechanical support is done byXylem vesselsXylem tracheidsXylem fibresBoth A and B
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.