You would like to use a pipe-based continuous flow hydroponic system. In this configuration, the roots of the strawberry plants are exposed to water that flows continuously through cylindrical pipes, driven by a pressure pump. The pipes are oriented horizontally so that the nutrient-containing water flows in the axial (x) direction. You are wondering whether this design is sufficient to support the needs of your strawberry plants. Let’s begin by considering the transport of nutrient-containing water through one hollow pipe. The viscosity of nutrient-containing water is 1.5 x 10-3 N·s/m2 and the density is 1000 kg/m3. Gravitational acceleration is g = 9.8 m/s2. The pipe has an inner diameter, D = 10 cm, and a length, L = 150 cm. The pressure at the inlet is Pin and the pressure at the outlet is Pout. a) Sketch the system, labeling all pressures, dimensions, and proportionality constants. b) Beginning with either the generalized momentum balance or the Navier-Stokes equations, derive an expression for the steady-state velocity profile of the water (vx) as a function of radius, r. Make sure to state all assumptions and boundary conditions. c) What pressure drop (Pin – Pout) is required to deliver 2 L/min of water through the pipe? You are worried that this rate of flow might damage the delicate roots of the strawberry plants. d) What is the shear stress on the roots of the plants, assuming they are located at the inner wall of the pipe (at r = 0.5D)? You are not willing to risk losing your strawberry harvest to root damage. As an alternative design, you consider packing the cylindrical pipe with spherical water beads (Orbeez®, with radius rbead = 0.5 cm, void fraction = 0.1) to provide structural support to the roots. e) Sketch the new system, labeling all pressures, dimensions, and proportionality constants. f) If the plants still require a volumetric flowrate of 2L/min of water,what pressure drop is required for the Orbeez®-packed pipe? Make sure to state all assumptions. g) What is the average velocity of the flow through the Orbeez®-packed pipe, ? How does this compare to the average velocity in the empty pipe? h) Will the presence of the Orbeez® decrease the shear stress on the roots? Explain your reasoning in 1- 2 sentences.
You would like to use a pipe-based continuous flow hydroponic system. In this configuration, the roots of the strawberry plants are exposed to water that flows continuously through cylindrical pipes, driven by a pressure pump. The pipes are oriented horizontally so that the nutrient-containing water flows in the axial (x) direction. You are wondering whether this design is sufficient to support the needs of your strawberry plants.
Let’s begin by considering the transport of nutrient-containing water through one hollow pipe. The viscosity of nutrient-containing water is 1.5 x 10-3 N·s/m2 and the density is 1000 kg/m3. Gravitational acceleration is g = 9.8 m/s2. The pipe has an inner diameter, D = 10 cm, and a length, L = 150 cm. The pressure at the inlet is Pin and the pressure at the outlet is Pout.
a) Sketch the system, labeling all pressures, dimensions, and proportionality constants.
b) Beginning with either the generalized momentum balance or the Navier-Stokes equations, derive an expression for the steady-state velocity profile of the water (vx) as a function of radius, r. Make sure to state all assumptions and boundary conditions.
c) What pressure drop (Pin – Pout) is required to deliver 2 L/min of water through the pipe?
You are worried that this rate of flow might damage the delicate roots of the strawberry plants.
d) What is the shear stress on the roots of the plants, assuming they are located at the inner wall of the pipe (at r = 0.5D)?
You are not willing to risk losing your strawberry harvest to root damage. As an alternative design, you consider packing the cylindrical pipe with spherical water beads (Orbeez®, with radius rbead = 0.5 cm, void fraction = 0.1) to provide structural support to the roots.
e) Sketch the new system, labeling all pressures, dimensions, and proportionality constants.
f) If the plants still require a volumetric flowrate of 2L/min of water,what pressure drop is required for the
Orbeez®-packed pipe? Make sure to state all assumptions.
g) What is the average velocity of the flow through the Orbeez®-packed pipe, <v>? How does this
compare to the average velocity in the empty pipe?
h) Will the presence of the Orbeez® decrease the shear stress on the roots? Explain your reasoning in 1- 2 sentences.
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