1. Draining a Sphere-Tube System A spherical tank of radius R is filled initially to a...

 

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1. Draining a Sphere-Tube System A spherical tank of radius R is filled initially to a height Ho. The tank drains from the bottom through a vertical tube of length I, and radius To. The tube radius is sufficiently small that the flow in the tube is laminar. a) Write the differential equation for H as a function of time, H(t). This equation contains the fluid density p. gravitational acceleration g, fluid viscosity μ, and the geometric quantities: R. ro, and lo HINT: In class we related height to flow rate via Toricelli's Law, in this problem due to the laminar flow in the tube we use Poiseulle's Law: YAP where Bulo b) Rewrite the differential equation in terms of dimensionless variables: Q(t)= A= f= c) Why is the definition in part (b) for f preferable to the following? pgr t 8μl R² IC AP=pg (H + lo) pgrt 8μR³ pgroteff 8μR³ Q d) Integrate the differential equation. HINT: easier with substitution of λ = A + Co e) Obtain the dimensionless efflux time for the sphere, teff in terms of Ho and l (where F. =) f) If the tank is initially full (so that F, = 2), what is the expression for feff as a function of ? g) Use Excel to plot your result for feff() over the range, 0 < < 10. k 20 1. Draining a Sphere-Tube System A spherical tank of radius R is filled initially to a height Ho. The tank drains from the bottom through a vertical tube of length I, and radius To. The tube radius is sufficiently small that the flow in the tube is laminar. a) Write the differential equation for H as a function of time, H(t). This equation contains the fluid density p. gravitational acceleration g, fluid viscosity μ, and the geometric quantities: R. ro, and lo HINT: In class we related height to flow rate via Toricelli's Law, in this problem due to the laminar flow in the tube we use Poiseulle's Law: YAP where Bulo b) Rewrite the differential equation in terms of dimensionless variables: Q(t)= A= f= c) Why is the definition in part (b) for f preferable to the following? pgr t 8μl R² IC AP=pg (H + lo) pgrt 8μR³ pgroteff 8μR³ Q d) Integrate the differential equation. HINT: easier with substitution of λ = A + Co e) Obtain the dimensionless efflux time for the sphere, teff in terms of Ho and l (where F. =) f) If the tank is initially full (so that F, = 2), what is the expression for feff as a function of ? g) Use Excel to plot your result for feff() over the range, 0 < < 10. k 20