# 3.28: Untitled Page 54

## Chapter 3

Part (c). Use the bisection method described in Example 3.5 nd a

Appendix B1 to

solve the governing equation in order to determine (

h t) for the following

conditions:

2

2

 0 010

.

cm s ,

g  980 cm s ,

r  0 010

.

cm ,

o

3

  70 dyne cm ,

  1 g cm

If a very fine capillary tube is available (

o

r

0 010

.

cm ), you can test your analysis

by doing a simple experiment in which the capillary rise is measured as a nction

fu

of time.

3‐18. In Figure 3.18 we have illustrated a cross‐sectional view of a barge loaded with stones. The barge

has sprung a lead as indicated, and the volumetric flow

rate of the leak is given by

Q

C A

o

(

g h h )

leak

d

i

Here C is a

ge

dischar

coefficient equal to 0.6,

d

o

A is the area of the hole through

hich

w

the water is

Figure 3.18 Leaking barge

leaking, h

is the height of the external water surface above the bottom of the barge, and h is the internal height of the water above the bottom of the barge.

i

The initial

conditions for this problem are

I.C.

h h , h

,

t

o

0

0

i

Single component systems

93

and you are asked to determine when the barge will sink. The length of the barge is L and the space available for water inside the barge is  HwL . Here  is usually referred to as the void fraction and for this particular load of stones

  0 . 35 .

In order to solve this problem you will need to make use of the fact that the buoyancy force acting on the barge is

buoyancy force  ( gh) wL

where  is the density of water. This buoyancy force is equal and opposite to the gravitational force acting on the barge, and this is given by gravitational force  mg

Here m represents the mass of the barge, the stones, and the water that has leaked into the barge. The amount of water that has leaked into the barge is given by

volume of water in the barge   h ( w )

i

L

Given the following parameters:

2

w  30 ft , L  120 ft , A

 0 . 03 ft , h  8 ft , H  12 ft

o

o

determine how long it will take before the barge sinks. You can compare your solution to this problem with an experiment done in your bathtub. Fill a coffee can with rocks and weigh it; then add water and weigh it again in order to determine the void fraction. Remove the water (but not the rocks) and drill a small hole in the bottom. Measure the diameter of the hole (it should be about 0.1 cm) so that you know the area of the leak, and place the can in a bathtub filled with water. Measure the time required for the can to sink.

3‐19. The solution to Problem 3‐9 indicates that the diversion of water from Mono Lake to Los Angeles would cause the level of the lake to drop 19 meters.

A key parameter in this prediction is the evaporation rate of 36 in/year, and the steady‐state analysis gave no indication of the time required for this reduction to occur. In this problem you are asked to develop the unsteady analysis of the Mono Lake water balance. Use available experimental data to predict the evaporation rate, and then use your solution and the new value of the evaporation rate to predict the final values of the radius and the depth of the lake. You are also asked to predict the number of years required for the

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