Problem 6.54*

Jewett, Walker, Reese, McMullan

Given

A swimming pool W by Y by h deep is to be emptied by gravity through a long pipe.

Figure 1

Given

Constant Name Constant Value Constant Name Constant Value

Coefficient of friction f Height h

Area of Bottom W by Y Length of Pipe L

Diameter of pipe D Roughness Value ε

Table 1

Problem

Derive a formula for the time to empty the tank from an initial level h0.

Assumptions

Constant average friction factor Liquid SteadyIncompressibleNeglect Minor LossesGage pressure.The area of the tank is much greater than the area of the pipe.Horizontal axis is at the bottom of the tank.

Solution

Start with the energy equation:

)1.1(22

2

2

221

2

11fhz

g

V

pg

Pz

g

V

pg

P

Therefore:

)2.1(2

2

221 fh

g

V

pg

Pz

Solution

)3.1(2 fpghP

)4.1(01 hz

So the energy equation becomes:

)5.1(2

2

20 f

f hg

V

pg

pghh

Assume:

Assume:

)6.1(2

22

gD

fLVh f

Solution

Flow in equals flow out:

Simplify equation 1.5:

Substitute into equation 1.7:

)7.1(2

2

22

20

g

V

gD

fLVh

)8.1(2

112

A

VAV

)9.1(2 2

2

21

2

122

21

21

0gA

VA

gDA

VfLAh

Solution

Isolate V1:

Factor out common terms:

Multiply equation 1.11 by ho2:

(1.10))2g

1

gD

fL(

A

VAh

22

21

21

0

)11.1()2

1(1220

21

21 ggD

fL

Ah

A

V

)12.1()2

( 0022

212

g

h

gD

fLh

A

At

Solution

Take the square root to get the final answer:

.)(2

00

2

1 ansg

h

gD

fLh

A

At

BME Application

An application of this problem in Biofluids is an IV drip because IVs are emptied using gravity. It is very important for doctors to know how much time it will take to empty an IV bag. The tubing that the IV is emptied into will have certain characteristics that will affect the flow. All of these considerations need to be taken into account in order to administer proper dosages.