3. A conical container has a half-angle a as shown above. Liquid is poured in at a constant rate Q (volume per time). Simultaneously, liquid leaks out at a rate q that is proportional to the current height of the liquid, i.e. q = kh. (a) Set up the differential equation governing h(t). (b) Neglecting leakage (k = 0), solve for h(t) given that the container is initially empty. How long does it take to fill the container up to a total height H? Estimate the answer first using scaling arguments in addition to obtain an exact solution. (c) Now imagine that the container is full up to a height H at t= 0 but no more fluid is poured in (Q = 0). Solve for h(t). How long does it take for the container to be drained by the leak? Estimate the drain time using scaling arguments, then also obtain the solution exactly.

3. A conical container has a half-angle a as shown above. Liquid is poured in at a constant rate Q (volume per time). Simultaneously, liquid leaks out at a rate q that is proportional to the current height of the liquid, i.e. q = kh. (a) Set up the differential equation governing h(t). (b) Neglecting leakage (k = 0), solve for h(t) given that the container is initially empty. How long does it take to fill the container up to a total height H? Estimate the answer first using scaling arguments in addition to obtain an exact solution. (c) Now imagine that the container is full up to a height H at t= 0 but no more fluid is poured in (Q = 0). Solve for h(t). How long does it take for the container to be drained by the leak? Estimate the drain time using scaling arguments, then also obtain the solution exactly.

Elements Of Electromagnetics

7th Edition

ISBN:9780190698614

Author:Sadiku, Matthew N. O.

Publisher:Sadiku, Matthew N. O.

ChapterMA: Math Assessment

Section: Chapter Questions

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Concept explainers

Fluid Kinematics

The branch of science that deals with objects which are either in motion or stationary are studied under the branch of classical mechanics. Fluid mechanics is a subcategory of classical mechanics that deals with the behavior of fluids at rest (fluid statics) or in motion (posing some velocity). In fluid kinematics, studies focus on the associated motion of fluid particles or groups of particles, and its associated parameters like displacements, velocity, and acceleration without concerning about the forces that caused the motion. All the laws of classical mechanics are applied in the study of fluid kinematics.

Question

3. A conical container has a half-angle a as shown above. Liquid is poured in at a constant rate
Q (volume per time). Simultaneously, liquid leaks out at a rate q that is proportional to the
current height of the liquid, i.e. qL kh.
=
(a) Set up the differential equation governing h(t).
=
(b) Neglecting leakage (k) 0), solve for h(t) given that the container is initially empty.
How long does it take to fill the container up to a total height H? Estimate the answer
first using scaling arguments in addition to obtain an exact solution.
(c) Now imagine that the container is full up to a height H at t = 0 but no more fluid
is poured in (Q = 0). Solve for h(t). How long does it take for the container to be
drained by the leak? Estimate the drain time using scaling arguments, then also obtain
the solution exactly.

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Step 1: Calculate the volume of the liquid in the container:

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Step 2: (a) Set up the differential equation governing h(t):

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Step 3: (b) Neglect leakage (k=0), Solve for h(t) given that container is initially empty:

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Step 4: (c) Estimate the time taken for the container to be drained by the leak:

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