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Problems

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Properties Related to Mass and Weight

2.1 * Describe how density differs from specific weight.

2.2 * For what fluids can we (usually) assume density to be nearly constant? For what fluids must we be

careful to calculate density as a function of temperature and pressure?

2.3 * Where in this text can you find density data for such fluids as oil and mercury?

2.4 An engineer living at an elevation of 2500 ft is conducting experiments to verify predictions of glider

performance. To process data, density of ambient air is needed. The engineer measures temperature

(74.3°F) and atmospheric pressure (27.3 inches of mercury). Calculate density in units of kg/m 3 . Compare

the calculated value with data from Table A.3 and make a recommendation about the effects of elevation on

density; that is, are the effects of elevation significant?

Answer:

Local conditions: ρ = 1.09 kg/m 3 ;

table value: ρ = 1.22 kg/m 3

2.5 Calculate the density and specific weight of carbon dioxide at a pressure of 300 kN/m 2 absolute and 60°C.

2.6 Determine the density and specific weight of methane gas at at a pressure of 300 kN/m 2 absolute and 60°C.

Answer:

2.7 Natural gas is stored in a spherical tank at a temperature of 10°C. At a given initial time, the pressure in the

tank is 100 kPa gage, and the atmospheric pressure is 100 kPa absolute. Some time later, after considerably

more gas is pumped into the tank, the pressure in the tank is 200 kPa gage, and the temperature is still 10°C.

What will be the ratio of the mass of natural gas in the tank when p = 200 kPa gage to that when the

pressure was 100 kPa gage?

2.8 At a temperature of 100°C and an absolute pressure of 5 atmospheres, what is the ratio of the density of

water to the density of air, ρ w /ρ a ?

Answer:

ρ water /ρ air = 203

2.9 Find the total weight of a 10 ft 3 tank of oxygen if the oxygen is pressurized to 500 psia, the tank itself

weighs 150 lbf, and the temperature is 70°F?

2.10 A 10 m 3 oxygen tank is at 15°C and 800 kPa. The valve is opened, and some oxygen is released until the

pressure in the tank drops to 600 kPa. Calculate the mass of oxygen that has been released from the tank if

the temperature in the tank does not change during the process.

Answer:

Mass released is 26.7 kg.

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2.11 What are the specific weight and density of air at an absolute pressure of 600 kPa and a temperature of

50°C?

2.12 Meteorologists often refer to air masses in forecasting the weather. Estimate the mass of 1 mi 3 of air in

slugs and kilograms. Make your own reasonable assumptions with respect to the conditions of the

atmosphere.

Answer:

2.13 A bicycle rider has several reasons to be interested in the effects of temperature on air density. The

aerodynamic drag force decreases linearly with density. Also, a change in temperature will affect the tire

pressure.

a. To visualize the effects of temperature on air density, write a computer program that calculates the

air density at atmospheric pressure for temperatures from 10°C to 50°C.

b. Also assume that a bicycle tire was inflated to an absolute pressure of 450 kPa at 20°C. Assume the

volume of the tire does not change with temperature. Write a program to show how the tire pressure

changes with temperature in the same temperature range, 10°C to 50°C.

Prepare a table or graph of your results for both problems. What engineering insights do you gain from

these calculations?

2.14 A design team is developing a prototype CO 2 cartridge for a manufacturer of rubber rafts.This cartridge

will allow a user to quickly inflate a raft. A typical raft is shown in the sketch. Assume a raft inflation

pressure of 3 psi (this means that the absolute pressure is 3 psi greater than local atmospheric pressure).

Estimate the volume of the raft and the mass of CO 2 in grams in the prototype cartridge.

PROBLEM 2.14

Answer:

2.15 A team is designing a heliumfilled balloon that will fly to an altitude of 80,000 ft. As the balloon ascends,

the upward force (buoyant force) will need to exceed the total weight. Thus, weight is critical. Estimate the

weight (in newtons) of the helium inside the balloon. The balloon is inflated at a site where the

atmospheric pressure is 0.89 bar and the temperature is 22°C. When inflated prior to launch, the balloon is

spherical (radius 1.3 m) and the inflation pressure equals the local atmospheric pressure.

2.16 Hydrometers are used in the wine and beer industries to measure the alcohol content of the product. This is

accomplished by measuring the specific gravity of the liquid before fermentation, during fermentation, or

after fermentation is complete. During fermentation, glucose (C 6 H 12 O 6 ) is converted to ethyl alcohol

(CH 3 CH 2 OH) and carbon dioxide gas, which escapes from the vat.

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Brewer's yeast tolerates alcohol contents to approximately 5% before fermentation stops, whereas wine

yeast tolerates alcohol contents up to 21% depending on the yeast strain. The specific gravity of alcohol is

0.80, and the maximum specific gravity of sugar in solution is 1.59. If a wine has a specific gravity of 1.08

before fermentation, and all the sugar is converted to alcohol, what will be the final specific gravity of the

wine and the percent alcohol content by volume? Assume that the initial liquid (the unfermented wine is

called must) contains only sugar and water.

Answer:

S f = 0.972, percent alcohol by volume = 13.7%

Viscosity

2.17 * The following questions relate to viscosity.

a. What are the primary dimensions of viscosity? What are five common units?

b. What is the viscosity of SAE 10W30 motor oil at 115°F (in traditional units)?

c. How does viscosity of water vary with temperature? Why?

d. How does viscosity of air vary with temperature? Why?

2.18 What is the change in the viscosity and density of water between 10°C and 70°C? What is the change in the

viscosity and density of air between 10°C and 70°C? Assume standard atmospheric pressure ( p = 101

kN/m 2 absolute).

Answer:

For water, = 9.06 × 10 4 N · s/m 2 and ρ = 22 kg/m 3 ; for air, = 2.8 × 10 6 N · s/m 2 and ρ = 0.22

kg/m 3 .

2.19 Determine the change in the kinematic viscosity of air that is heated from 10°C to 70°C. Assume standard

atmospheric pressure.

2.20 Find the dynamic and kinematic viscosities of kerosene, SAE 10W30 motor oil, and water at a

temperature of 38°C (100°F).

Answer:

For oil, = 6.7 × 10 2 N · s/m 2 and ν = 7.6 × 10 5 m 2 /s; for kerosene, = 1.4 × 10 3 N · s/m 2 and

ν = 1.7 × 10 6 m 2 /s; for water, = 6.8 × 10 4 N · s/m 2 and ν = 6.8 × 10 7 m 2 /s

2.21 What is the ratio of the dynamic viscosity of air to that of water at standard pressure and a temperature of

20°C? What is the ratio of the kinematic viscosity of air to that of water for the same conditions?

2.22 Using Sutherland's equation and the ideal gas law, develop an expression for the kinematic viscosity ratio

ν/ν 0 in terms of pressures p and p 0 and temperatures T and T 0 , where the subscript 0 refers to a reference

condition.

Answer:

ν/ν 0 = ( p 0 / p )( T / T 0 ) 5/2 ( T 0 + S)/( T + S )

2.23 The dynamic viscosity of air at 15°C is 1.78 × 10 5 N · s/m 2 . Using Sutherland's equation, find the viscosity

at 100°C.

2.24 The kinematic viscosity of methane at 15°C and atmospheric pressure is 1.59 × 10 5 m 2 /s. Using

Sutherland's equation and the ideal gas law, find the kinematic viscosity at 200°C and 2 atmospheres.

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Answer:

ν = 1.99 × 10 5 m 2 /s

2.25 The dynamic viscosity of nitrogen at 59°F is 3.59 × 10 7 lbf · s/ft 2 . Using Sutherland's equation, find the

dynamic viscosity at 200°F.

2.26 The kinematic viscosity of helium at 59°F and 1 atmosphere is 1.22 × 10 3 ft 2 /s. Using Sutherland's

equation and the ideal gas law, find the kinematic viscosity at 30°F and a pressure of 1.5 atmospheres.

Answer:

ν = 1.66 × 10 3 ft 2 /s

2.27 The absolute viscosity of propane at 100°C is 1.00 × 10 5 N · s/m 2 and at 400°C is 1.72 × 10 5 N · s/m 2 .

Find Sutherland's constant for propane.

2.28 Ammonia is very volatile, so it may be either a gas or a liquid at room temperature. When it is a gas, its

absolute viscosity at 68°F is 2.07 × 10 7 lbf · s/ft 2 and at 392°F is 3.46 × 10 7 lbf · s/ft 2 . Using these two

data points, find Sutherland's constant for ammonia.

Answer:

S = 903°R

2.29 The viscosity of SAE 10W30 motor oil at 38°C is 0.067 N · s/m 2 and at 99°C is 0.011 N · s/m 2 . Using Eq.

2.8 for interpolation, find the viscosity at 60°C. Compare this value with that obtained by linear

interpolation.

2.30 The viscosity of grade 100 aviation oil at 100°F is 4.43 × 10 3 lbf · s/ft 2 and at 210°F is 3.9 × 10 4 lbf · s/ft 2 .

Using Eq. 2.8, find the viscosity at 150°F.

Answer:

= 1.32 × 10 3 lbfs/ft 2

2.31 Two plates are separated by a 1/8in. space. The lower plate is stationary; the upper plate moves at a

velocity of 25 ft/s. Oil (SAE 10W30, 150°F), which fills the space between the plates, has the same

velocity as the plates at the surface of contact. The variation in velocity of the oil is linear. What is the

shear stress in the oil?

2.32 Find the kinematic and dynamic viscosities of air and water at a temperature of 40°C (104°F) and an

absolute pressure of 170 kPa (25 psia).

Answer:

For air, air = 1.91 × 10 5 N · s/m 2 and ν air = 10.1 × 10 6 m 2 /s; for water, water = 6.53 × 10 5 N · s/m 2 and

ν water = 6.58 × 10 7 m 2 /s.

2.33 The sliding plate viscometer shown below is used to measure the viscosity of a fluid. The top plate is

moving to the right with a constant velocity of 10 m/s in response to a force of 3 N. The bottom plate is

stationary. What is the viscosity of the fluid? Assume a linear velocity distribution.

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PROBLEM 2.33

2.34 The velocity distribution for water (20°C) near a wall is given by u = a ( y / b ) 1/6 , where a = 10 m/s,

b = 2 mm, and y is the distance from the wall in mm. Determine the shear stress in the water at y = 1 mm.

PROBLEM 2.35, 2.36 and 2.37

Answer:

τ( y = 1 mm) = 1.49 Pa

2.35 The velocity distribution for the flow of crude oil at 100°F ( = 8 × 10 5 lbf · s/ft 2 ) between two walls is

shown, and is given by u = 100 y (0.1 y ) ft/s, where y is measured in feet and the space between the walls

is 0.1 ft. Plot the velocity distribution and determine the shear stress at the walls.

2.36 A liquid flows between parallel boundaries as shown above. The velocity distribution near the lower wall

is given in the following table:

Y in mm V in m/s

0.0 0.00

1.0 1.00

2.0 1.99

3.0 2.98

a. If the viscosity of the liquid is 10 3 N · s/m 2 , what is the maximum shear stress in the liquid?

b. Where will the minimum shear stress occur?

Answer:

a. τ max = 1.0 N/m 2 ;

b. minimum shear stress will occur midway between the two walls

2.37 Suppose that glycerin is flowing (T = 20°C) and that the pressure gradient dp / dx is 1.6 kN/m 3 . What are

the velocity and shear stress at a distance of 12 mm from the wall if the space B between the walls is

5.0 cm? What are the shear stress and velocity at the wall? The velocity distribution for viscous flow

between stationary plates is

2.38 A laminar flow occurs between two horizontal parallel plates under a pressure gradient dp / ds ( p decreases

in the positive s direction). The upper plate moves left (negative) at velocity u t . The expression for local

velocity u is given as

a. Is the magnitude of the shear stress greater at the moving plate ( y = H ) or at the stationary plate

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