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Suppose that exactly half of the annular volume between concentric, horizontal cylinders is filled with a liquid and half with a gas, as shown on the top left corner of the next page. When the inner cylinder is rotated at an angular velocity, it is found that one interface rises to a height H above the other. The gap between the cylinders is sufficiently thin (W << R) that local Cartesian coordinates can be used, as shown on the right. (For clarity, the gap width is exaggerated on the left.) All of the following questions concern flow in the liquid, which is Newtonian with constant properties. g R+W Gas Ω Liquid R Enlargement of liquid-filled gap y X W ΩR Fixed (a) Assuming steady, fully developed flow, determine Vx(y) in the liquid in terms of dP/dx. (At this point, dP/dx is unknown.) (b) Relate dIP/dx in the liquid to H. You can assume that the absolute pressure in the gas is nearly constant (P = Po) and that surface tension is negligible. (c) Calculate H. That is, relate H to the rotation rate (), cylinder radius (R), gap width (W), liquid viscosity (µ) and liquid density (p).

Suppose that exactly half of the annular volume between concentric, horizontal cylinders is filled with a liquid and half with a gas, as shown on the top left corner of the next page. When the inner cylinder is rotated at an angular velocity, it is found that one interface rises to a height H above the other. The gap between the cylinders is sufficiently thin (W << R) that local Cartesian coordinates can be used, as shown on the right. (For clarity, the gap width is exaggerated on the left.) All of the following questions concern flow in the liquid, which is Newtonian with constant properties. g R+W Gas Ω Liquid R Enlargement of liquid-filled gap y X W ΩR Fixed (a) Assuming steady, fully developed flow, determine Vx(y) in the liquid in terms of dP/dx. (At this point, dP/dx is unknown.) (b) Relate dIP/dx in the liquid to H. You can assume that the absolute pressure in the gas is nearly constant (P = Po) and that surface tension is negligible. (c) Calculate H. That is, relate H to the rotation rate (), cylinder radius (R), gap width (W), liquid viscosity (µ) and liquid density (p).

Introduction to Chemical Engineering Thermodynamics
Introduction to Chemical Engineering Thermodynamics
8th Edition
ISBN:9781259696527
Author:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Publisher:J.M. Smith Termodinamica en ingenieria quimica, Hendrick C Van Ness, Michael Abbott, Mark Swihart
Chapter1: Introduction
Section: Chapter Questions
Problem 1.1P
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Suppose that exactly half of the annular volume between concentric, horizontal cylinders is filled with a liquid and half with a gas, as shown on the top left corner of the next page. When the inner cylinder is rotated at an angular velocity , it is found that one interface rises to a height H above the other. The gap between the cylinders is sufficiently thin (W << R) that local Cartesian coordinates can be used, as shown on the right. (For clarity, the gap width is exaggerated on the left.) All of the following questions concern flow in the liquid, which is Newtonian with constant properties. R+W Gas Liquidi R H y Enlargement of liquid-filled gap X W ΩR Fixed (a) Assuming steady, fully developed flow, determine Vx(y) in the liquid in terms of dp/dx. (At this point, dP/dx is unknown.) (b) Relate dIP/dx in the liquid to H. You can assume that the absolute pressure in the gas is nearly constant (P = Po) and that surface tension is negligible. (c) Calculate H. That is, relate H to the rotation rate (2), cylinder radius (R), gap width (W), liquid viscosity (u) and liquid density (p).
Transcribed Image Text:Suppose that exactly half of the annular volume between concentric, horizontal cylinders is filled with a liquid and half with a gas, as shown on the top left corner of the next page. When the inner cylinder is rotated at an angular velocity , it is found that one interface rises to a height H above the other. The gap between the cylinders is sufficiently thin (W << R) that local Cartesian coordinates can be used, as shown on the right. (For clarity, the gap width is exaggerated on the left.) All of the following questions concern flow in the liquid, which is Newtonian with constant properties. R+W Gas Liquidi R H y Enlargement of liquid-filled gap X W ΩR Fixed (a) Assuming steady, fully developed flow, determine Vx(y) in the liquid in terms of dp/dx. (At this point, dP/dx is unknown.) (b) Relate dIP/dx in the liquid to H. You can assume that the absolute pressure in the gas is nearly constant (P = Po) and that surface tension is negligible. (c) Calculate H. That is, relate H to the rotation rate (2), cylinder radius (R), gap width (W), liquid viscosity (u) and liquid density (p).
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