A brand of toothpaste is made using a mixture of white and red paste. During the manufacturing process, both pastes are pumped through a pipe, with the red paste lying in the centre of the pipe, as shown in Figure Q2. The pipe has an internal radius, R₂, and the segment of red paste has a radius R₁. The flow is found to follow the Poiseuille equation, i.e. u(r): == 1 dp 4u dx (R²-r²) dp is the pressure gradient. The total flow of dx Where u is the viscosity of the paste and μ toothpaste through the pipe is Qtot, and the flowrate of the red paste is Qred. The toothpaste has a density of p = 1200 kg/m³, and a viscosity of 0.03 Pa.s. The pipe has a radius R₂ = 5 cm and a length L = 2 m. a) Find an expression for the flowrate of the red toothpaste. You answer should be in terms of µ, R₁, R₂, dx' b) The flow in the pipe becomes turbulent when the Reynolds number reaches 2000 (where Re is calculated using the mean velocity across the pipe). What is the maximum pressure drop along the pipe that can be achieved without the flow becoming turbulent? c) Engineers find that when the shear stress at the interface between the red and white paste exceeds 0.5 Pa the colours begin to mix. If R₁ = 2 cm, what is the maximum pressure drop along the pipe that can be achieved without the colours smudging?

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

Problem 1.1MA

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Transcribed Image Text:A brand of toothpaste is made using a mixture of white and red paste. During the manufacturing process, both pastes are pumped through a pipe, with the red paste lying in the centre of the pipe, as shown in Figure Q2. The pipe has an internal radius, R₂, and the segment of red paste has a radius R₁. The flow is found to follow the Poiseuille equation, i.e. u(r) = 1 dp - (R² - r²) dp Where u is the viscosity of the paste and μ is the pressure gradient. The total flow of toothpaste through the pipe is Qtot, and the flowrate of the red paste is Qred. dx 4μ dx The toothpaste has a density of p = 1200 kg/m³, and a viscosity of 0.03 Pa.s. The pipe has a radius R₂ = 5 cm and a length L = 2 m. R₁ a) Find an expression for the flowrate of the red toothpaste. You answer should be in terms of µ, R₁, R₂, P) b) The flow in the pipe becomes turbulent when the Reynolds number reaches 2000 (where Re is calculated using the mean velocity across the pipe). What is the maximum pressure drop along the pipe that can be achieved without the flow becoming turbulent? R₂ c) Engineers find that when the shear stress at the interface between the red and white paste exceeds 0.5 Pa the colours begin to mix. If R₁ = 2 cm, what is the maximum pressure drop along the pipe that can be achieved without the colours smudging? u(y) Figure Q2: Flow of toothpaste of two different colours.

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