A balloon is surrounded by a tank of liquid at negative pressure and is con- nected to the atmosphere by a tube of length L and cross-sectional area A (Fig. 7.25). The pressure inside the balloon p oscillates above and below atmospheric pressure causing small changes in the balloon volume V. The elasticity of the balloon is characterized by its compliance C, defined by Ap = AV/C (a) Derive a second-order differential equation for AV(t), assuming that (i) the pressure differential along the tube accelerates the air in the tube and is not used to overcome entrance, exit, or tube losses; and (ii) the air density p is constant. From the equation, show that the natural frequency of the system is JA/pLC. Atmosphere Length = L; Cross-sectional area = A Air at pressure p Liquid Balloon Figure 7.25

A balloon is surrounded by a tank of liquid at negative pressure and is con- nected to the atmosphere by a tube of length L and cross-sectional area A (Fig. 7.25). The pressure inside the balloon p oscillates above and below atmospheric pressure causing small changes in the balloon volume V. The elasticity of the balloon is characterized by its compliance C, defined by Ap = AV/C (a) Derive a second-order differential equation for AV(t), assuming that (i) the pressure differential along the tube accelerates the air in the tube and is not used to overcome entrance, exit, or tube losses; and (ii) the air density p is constant. From the equation, show that the natural frequency of the system is JA/pLC. Atmosphere Length = L; Cross-sectional area = A Air at pressure p Liquid Balloon Figure 7.25

Principles of Heat Transfer (Activate Learning with these NEW titles from Engineering!)

8th Edition

ISBN:9781305387102

Author:Kreith, Frank; Manglik, Raj M.

Publisher:Kreith, Frank; Manglik, Raj M.

Chapter5: Analysis Of Convection Heat Transfer

Section: Chapter Questions

Problem 5.5P: Evaluate the dimensionless groups hcD/k,UD/, and cp/k for water, n-butyl alcohol, mercury, hydrogen,...

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Question

A balloon is surrounded by a tank of liquid at negative pressure and is con-
nected to the atmosphere by a tube of length L and cross-sectional area A
(Fig. 7.25). The pressure inside the balloon p oscillates above and below
atmospheric pressure causing small changes in the balloon volume V. The
elasticity of the balloon is characterized by its compliance C, defined by
Ap = AV/C
(a) Derive a second-order differential equation for AV(t), assuming that
(i) the pressure differential along the tube accelerates the air in the tube
and is not used to overcome entrance, exit, or tube losses; and (ii) the air
density p is constant. From the equation, show that the natural frequency
of the system is JA/pLC.

Atmosphere
Length = L; Cross-sectional area = A
Air at
pressure p
Liquid
Balloon
Figure 7.25

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