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An experiment designed to measure the size of air bubbles released in water is diagrammed below. Air is pushed gently through a small tube of diameter d by a syringe. A bubble which forms at the tip releases when it reaches a critical diameter D, after which it soon reaches a terminal velocity U. Eventually, the bubble comes to rest against a microbalance that measures the upward force F exerted on it by the now-stationary bubble. The figure is not to scale; actually, D is at least ten times d. Assume that water has pw = 10³ kg/m³ and µw = 10-³ Pa·s. Also, assume that air has Pa = 1.2 kg/m³ and μa = 1.8×10-5 Pa.s. Micro- balance of Air supply U10 Bubble at rest Bubble rising Bubble near release (a) Obtain an expression which relates F to D. = (b) If an air bubble is very small and the water extremely pure, the drag is FD 2πμⓇDU, where µw is the viscosity of water. Show how to calculate U under these conditions. (c) The drag expression in part (b) requires that the Reynolds numbers based on air properties (Rea) and water properties (Rew) both be < 1. What is the maximum bubble size (in µm) for which those constraints hold? )

An experiment designed to measure the size of air bubbles released in water is diagrammed below. Air is pushed gently through a small tube of diameter d by a syringe. A bubble which forms at the tip releases when it reaches a critical diameter D, after which it soon reaches a terminal velocity U. Eventually, the bubble comes to rest against a microbalance that measures the upward force F exerted on it by the now-stationary bubble. The figure is not to scale; actually, D is at least ten times d. Assume that water has pw = 10³ kg/m³ and µw = 10-³ Pa·s. Also, assume that air has Pa = 1.2 kg/m³ and μa = 1.8×10-5 Pa.s. Micro- balance of Air supply U10 Bubble at rest Bubble rising Bubble near release (a) Obtain an expression which relates F to D. = (b) If an air bubble is very small and the water extremely pure, the drag is FD 2πμⓇDU, where µw is the viscosity of water. Show how to calculate U under these conditions. (c) The drag expression in part (b) requires that the Reynolds numbers based on air properties (Rea) and water properties (Rew) both be < 1. What is the maximum bubble size (in µm) for which those constraints hold? )

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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An experiment designed to measure the size of air bubbles released in water is diagrammed below. Air is pushed gently through a small tube of diameter d by a syringe. A bubble which forms at the tip releases when it reaches a critical diameter D, after which it soon reaches a terminal velocity U. Eventually, the bubble comes to rest against a microbalance that measures the upward force F exerted on it by the now-stationary bubble. The figure is not to scale; actually, D is at least ten times d. Assume that water has pw = 10³ kg/m³ and μw = 10-³ Pa·s. Also, assume that air has Pa = 1.2 kg/m³ and μa = 1.8x10-5Pa.s. Micro- balance g Air supply Bubble at rest Bubble O rising Bubble near release (a) Obtain an expression which relates F to D. (b) If an air bubble is very small and the water extremely pure, the drag is FD 2μⓇDU, where µw is the viscosity of water. Show how to calculate U under these conditions. (c) The drag expression in part (b) requires that the Reynolds numbers based on air properties (Rea) and water properties (Rew) both be < 1. What is the maximum bubble size (in µm) for which those constraints hold?
Transcribed Image Text:An experiment designed to measure the size of air bubbles released in water is diagrammed below. Air is pushed gently through a small tube of diameter d by a syringe. A bubble which forms at the tip releases when it reaches a critical diameter D, after which it soon reaches a terminal velocity U. Eventually, the bubble comes to rest against a microbalance that measures the upward force F exerted on it by the now-stationary bubble. The figure is not to scale; actually, D is at least ten times d. Assume that water has pw = 10³ kg/m³ and μw = 10-³ Pa·s. Also, assume that air has Pa = 1.2 kg/m³ and μa = 1.8x10-5Pa.s. Micro- balance g Air supply Bubble at rest Bubble O rising Bubble near release (a) Obtain an expression which relates F to D. (b) If an air bubble is very small and the water extremely pure, the drag is FD 2μⓇDU, where µw is the viscosity of water. Show how to calculate U under these conditions. (c) The drag expression in part (b) requires that the Reynolds numbers based on air properties (Rea) and water properties (Rew) both be < 1. What is the maximum bubble size (in µm) for which those constraints hold?
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