1 1-1
Chapter INTRODUCTION AND BASIC CONCEPTS
Thermodynamics
1-1C Classical thermodynamics is based on experimental observations whereas statistical thermodynamics is based on the average behavior of large groups of particles.
1-2C On a downhill road the potential energy of the bicyclist is being converted to kinetic energy, and thus the bicyclist picks up speed. There is no creation of energy, and thus no violation of the conservation of energy principle.
1-3C There is no truth to his claim. It violates the second law of thermodynamics.
1-4C A car going uphill without the engine running would increase the energy of the car, and thus it would be a violation of the first law of thermodynamics. Therefore, this cannot happen. Using a level meter (a device with an air bubble between two marks of a horizontal water tube) it can shown that the road that looks uphill to the eye is actually downhill.
Mass, Force, and Units
1-5C Pound-mass lbm is the mass unit in English system whereas pound-force lbf is the force unit. One pound-force is the force required to accelerate a mass of 32.174 lbm by 1 ft/s 2. In other words, the weight of a 1-lbm mass at sea level is 1 lbf.
1-6C In this unit, the word light refers to the speed of light. The light-year unit is then the product of a velocity and time. Hence, this product forms a distance dimension and unit.
1-7C There is no acceleration, thus the net force is zero in both cases.
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1-2
1-8E The weight of a man on earth is given. His weight on the moon is to be determined. Analysis Applying Newton's second law to the weight force gives
W mg m W g 180 lbf 32.10 ft/s 2 32.174 lbm ft/s 2 1 lbf 180.4 lbm
Mass is invariant and the man will have the same mass on the moon. Then, his weight on the moon will be
W mg (180.4 lbm )(5.47 ft/s 2 ) 1 lbf 32.174 lbm ft/s 2 30.7 lbf
1-9 The interior dimensions of a room are given. The mass and weight of the air in the room are to be determined. Assumptions The density of air is constant throughout the room. Properties The density of air is given to be Analysis The mass of the air in the room is
m
= 1.16 kg/m3. ROOM AIR 6X6X8 m3
V
(1.16 kg/m 3 )(6 6 8 m 3 ) 334.1 kg
Thus,
W mg (334.1 kg)(9.81 m/s 2 ) 1N 1 kg m/s 2 3277 N
1-10 The variation of gravitational acceleration above the sea level is given as a function of altitude. The height at which the weight of a body will decrease by 1% is to be determined. z Analysis The weight of a body at the elevation z can be expressed as
W
In our case,
W
mg
m(9.807 332 10 6 z) .
0.99Ws
0.99mgs
0.99(m)(9.807)
Substituting,
0.99(9.81) (9.81 3.32 10
6
0
z) z 29,539 m
Sea level
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1-3
1-11E The mass of an object is given. Its weight is to be determined. Analysis Applying Newton's second law, the weight is determined to be
W mg (10 lbm )(32.0 ft/s 2 ) 1 lbf 32.174 lbm ft/s 2 9.95 lbf
1-12 The acceleration of an aircraft is given in g's. The net upward force acting on a man in the aircraft is to be determined. Analysis From the Newton's second law, the force applied is
F ma m(6 g ) (90 kg)(6 9.81 m/s 2 ) 1N 1 kg m/s 2 5297 N
1-13 A rock is thrown upward with a specified force. The acceleration of the rock is to be determined. Analysis The weight of the rock is
W mg (5 kg)(9.79 m/s 2 ) 1N 1 kg m/s 2 48.95 N
Then the net force that acts on the rock is
Fnet Fup Fdown 150 48.95 101.05 N
Stone
From the Newton's second law, the acceleration of the rock becomes
a F m 101.05 N 1 kg m/s 2 5 kg 1N 20.2 m/s 2
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1-4
1-14 EES Problem 1-13 is reconsidered. The entire EES solution is to be printed out, including the numerical results with proper units. Analysis The problem is solved using EES, and the solution is given below. W=m*g "[N]" m=5 [kg] g=9.79 [m/s^2] "The force balance on the rock yields the net force acting on the rock as" F_net = F_up - F_down"[N]" F_up=150 [N] F_down=W"[N]" "The acceleration of the rock is determined from Newton's second law." F_net=a*m "To Run the program, press F2 or click on the calculator icon from the Calculate menu" SOLUTION a=20.21 [m/s^2] F_down=48.95 [N] F_net=101.1 [N] F_up=150 [N] g=9.79 [m/s^2] m=5 [kg] W=48.95 [N]
1-15 Gravitational acceleration g and thus the weight of bodies decreases with increasing elevation. The percent reduction in the weight of an airplane cruising at 13,000 m is to be determined. Properties The gravitational acceleration g is given to be 9.807 m/s2 at sea level and 9.767 m/s2 at an altitude of 13,000 m. Analysis Weight is proportional to the gravitational acceleration g, and thus the percent reduction in weight is equivalent to the percent reduction in the gravitational acceleration, which is determined from
%Reduction in weight %Reduction in g g 100 g 9.807 9.767 100 9.807 0.41%
Therefore, the airplane and the people in it will weight 0.41% less at 13,000 m altitude. Discussion Note that the weight loss at cruising altitudes is negligible.
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1-5
Systems, Properties, State, and Processes
1-16C This system is a region of space or open system in that mass such as air and food can cross its control boundary. The system can also interact with the surroundings by exchanging heat and work across its control boundary. By tracking these interactions, we can determine the energy conversion characteristics of this system.
1-17C The system is taken as the air contained in the piston-cylinder device. This system is a closed or fixed mass system since no mass enters or leaves it.
1-18C Carbon dioxide is generated by the combustion of fuel in the engine. Any system selected for this analysis must include the fuel and air while it is undergoing combustion. The volume that contains this air-fuel mixture within piston-cylinder device can be used for this purpose. One can also place the entire engine in a control boundary and trace the system-surroundings interactions to determine the rate at which the engine generates carbon dioxide.
1-19C When analyzing the control volume selected, we must account for all forms of water entering and leaving the control volume. This includes all streams entering or leaving the lake, any rain falling on the lake, any water evaporated to the air above the lake, any seepage to the underground earth, and any springs that may be feeding water to the lake.
1-20C Intensive properties do not depend on the size (extent) of the system but extensive properties do.
1-21C The original specific weight is
W
1
V
If we were to divide the system into two halves, each half weighs W/2 and occupies a volume of V/2. The specific weight of one of these halves is
W /2 V /2
1
which is the same as the original specific weight. Hence, specific weight is an intensive property.
1-22C The number of moles of a substance in a system is directly proportional to the number of atomic particles contained in the system. If we divide a system into smaller portions, each portion will contain fewer atomic particles than the original system. The number of moles is therefore an extensive property.
1-23C For a system to be in thermodynamic equilibrium, the temperature has to be the same throughout but the pressure does not. However, there should be no unbalanced pressure forces present. The increasing pressure with depth in a fluid, for example, should be balanced by increasing weight.
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1-6
1-24C A process during which a system remains almost in equilibrium at all times is called a quasiequilibrium process. Many engineering processes can be approximated as being quasi-equilibrium. The work output of a device is maximum and the work input to a device is minimum when quasi-equilibrium processes are used instead of nonquasi-equilibrium processes.
1-25C A process during which the temperature remains constant is called isothermal; a process during which the pressure remains constant is called isobaric; and a process during which the volume remains constant is called isochoric.
1-26C The state of a simple compressible system is completely specified by two independent, intensive properties.
1-27C In order to describe the state of the air, we need to know the value of all its properties. Pressure, temperature, and water content (i.e., relative humidity or dew point temperature) are commonly cited by weather forecasters. But, other properties like wind speed and chemical composition (i.e., pollen count and smog index, for example} are also important under certain circumstances. Assuming that the air composition and velocity do not change and that no pressure front motion occurs during the day, the warming process is one of constant pressure (i.e., isobaric).
1-28C A process is said to be steady-flow if it involves no changes with time anywhere within the system or at the system boundaries.
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1-7
1-29 EES The variation of density of atmospheric air with elevation is given in tabular form. A relation for the variation of density with elevation is to be obtained, the density at 7 km elevation is to be calculated, and the mass of the atmosphere using the correlation is to be estimated. Assumptions 1 Atmospheric air behaves as an ideal gas. 2 The earth is perfectly sphere with a radius of 6377 km, and the thickness of the atmosphere is 25 km. Properties The density data are given in tabular form as z, km 0 1 2 3 4 5 6 8 10 15 20 25 , kg/m3 1.225 1.112 1.007 0.9093 0.8194 0.7364 0.6601 0.5258 0.4135 0.1948 0.08891 0.04008
r, km 6377 6378 6379 6380 6381 6382 6383 6385 6387 6392 6397 6402
Analysis Using EES, (1) Define a trivial function rho= a+z in equation window, (2) select new parametric table from Tables, and type the data in a two-column table, (3) select Plot and plot the data, and (4) select plot and click on "curve fit" to get curve fit window. Then specify 2 nd order polynomial and enter/edit equation. The results are: (z) = a + bz + cz2 = 1.20252 0.101674z + 0.0022375z2 for the unit of kg/m3,
(or, (z) = (1.20252 0.101674z + 0.0022375z2) 109 for the unit of kg/km3) where z is the vertical distance from the earth surface at sea level. At z = 7 km, the equation would give = 0.60 kg/m3. (b) The mass of atmosphere can be evaluated by integration to be
h
m
V
dV
z 0
(a bz cz 2 )4 (r0
z ) 2 dz
h
4
z 0
(a bz cz 2 )(r02
2r0 z z 2 )dz
4 ar02 h r0 (2a br0 )h 2 / 2 (a 2br0
cr02 )h 3 / 3 (b 2cr0 )h 4 / 4 ch 5 / 5
where r0 = 6377 km is the radius of the earth, h = 25 km is the thickness of the atmosphere, and a = 1.20252, b = -0.101674, and c = 0.0022375 are the constants in the density function. Substituting and multiplying by the factor 109 for the density unity kg/km3, the mass of the atmosphere is determined to be m = 5.092 1018 kg Discussion Performing the analysis with excel would yield exactly the same results. EES Solution for final result: a=1.2025166; b=-0.10167 c=0.0022375; r=6377; h=25 m=4*pi*(a*r^2*h+r*(2*a+b*r)*h^2/2+(a+2*b*r+c*r^2)*h^3/3+(b+2*c*r)*h^4/4+c*h^5/5)*1E+9
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1-8
Temperature
1-30C The zeroth law of thermodynamics states that two bodies are in thermal equilibrium if both have the same temperature reading, even if they are not in contact.
1-31C They are Celsius ( C) and kelvin (K) in the SI, and fahrenheit ( F) and rankine (R) in the English system.
1-32C Probably, but not necessarily. The operation of these two thermometers is based on the thermal expansion of a fluid. If the thermal expansion coefficients of both fluids vary linearly with temperature, then both fluids will expand at the same rate with temperature, and both thermometers will always give identical readings. Otherwise, the two readings may deviate.
1-33 A temperature is given in C. It is to be expressed in K. Analysis The Kelvin scale is related to Celsius scale by T(K] = T( C) + 273 Thus, T(K] = 37 C + 273 = 310 K
1-34E A temperature is given in C. It is to be expressed in F, K, and R. Analysis Using the conversion relations between the various temperature scales, T(K] = T( C) + 273 = 18 C + 273 = 291 K T( F] = 1.8T( C) + 32 = (1.8)(18) + 32 = 64.4 F T(R] = T( F) + 460 = 64.4 + 460 = 524.4 R
1-35 A temperature change is given in C. It is to be expressed in K. Analysis This problem deals with temperature changes, which are identical in Kelvin and Celsius scales. Thus, T(K] = T( C) = 15 K
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1-9
1-36E The temperature of steam given in K unit is to be converted to F unit. Analysis Using the conversion relations between the various temperature scales,
T ( C) T (K) 273 300 273 27 C T ( F) 1.8T ( C) 32 (1.8)(27) 32 80.6 F
1-37E The temperature of oil given in F unit is to be converted to C unit. Analysis Using the conversion relation between the temperature scales,
T ( C) T ( F) 32 1. 8 150 32 1. 8 65.6 C
1-38E The temperature of air given in C unit is to be converted to F unit. Analysis Using the conversion relation between the temperature scales,
T ( F) 1.8T ( C) 32 (1.8)(150) 32 302 F
1-39E A temperature range given in F unit is to be converted to C unit and the temperature difference in F is to be expressed in K, C, and R. Analysis The lower and upper limits of comfort range in C are
T ( C) T ( F) 32 1 .8 65 32 1 .8 18.3 C
T ( C)
T ( F) 32 1 .8
75 32 1 .8
23.9 C
A temperature change of 10 F in various units are
T (R ) T ( C) T (K ) T ( F) 10 R T ( F) 10 5.6 C 1.8 1.8 T ( C) 5.6 K
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1-10
Pressure, Manometer, and Barometer
1-40C The pressure relative to the atmospheric pressure is called the gage pressure, and the pressure relative to an absolute vacuum is called absolute pressure. 1-41C The blood vessels are more restricted when the arm is parallel to the body than when the arm is perpendicular to the body. For a constant volume of blood to be discharged by the heart, the blood pressure must increase to overcome the increased resistance to flow.
1-42C No, the absolute pressure in a liquid of constant density does not double when the depth is doubled. It is the gage pressure that doubles when the depth is doubled.
1-43C If the lengths of the sides of the tiny cube suspended in water by a string are very small, the magnitudes of the pressures on all sides of the cube will be the same.
1-44C Pascal's principle states that the pressure applied to a confined fluid increases the pressure throughout by the same amount. This is a consequence of the pressure in a fluid remaining constant in the horizontal direction. An example of Pascal's principle is the operation of the hydraulic car jack.
1-45E The maximum pressure of a tire is given in English units. It is to be converted to SI units. Assumptions The listed pressure is gage pressure. Analysis Noting that 1 atm = 101.3 kPa = 14.7 psi, the listed maximum pressure can be expressed in SI units as
Pmax
35 psi (35 psi)
101.3 kPa 14.7 psi
241 kPa
Discussion We could also solve this problem by using the conversion factor 1 psi = 6.895 kPa.
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1-11
1-46 The pressure in a tank is given. The tank's pressure in various units are to be determined. Analysis Using appropriate conversion factors, we obtain (a)
P (1500 kPa ) 1 kN/m 2 1 kPa 1 kN/m 2 1 kPa 1 kN/m 2 1 kPa 1500 kN/m 2 1000 kg m/s 2 1 kN 1000 kg m/s 2 1 kN
(b)
P
(1500 kPa )
1,500,000 kg/m s 2
(c)
P
(1500 kPa )
1000 m 1 km
1,500,000,000 kg/km s 2
1-47E The pressure given in kPa unit is to be converted to psia. Analysis Using the kPa to psia units conversion factor,
P (200 kPa ) 1 psia 6.895 kPa 29.0 psia
1-48E A manometer measures a pressure difference as inches of water. This is to be expressed in psia unit. Properties The density of water is taken to be 62.4 lbm/ft 3 (Table A-3E). Analysis Applying the hydrostatic equation,
P gh (62.4 lbm/ft 3 )(32.174 ft/s 2 )(40/12 ft) 1.44 lbf/in 2 1.44 psia 1 lbf 32.174 lbm ft/s 2 1 ft 2 144 in 2
1-49 The pressure given in mm Hg unit is to be converted to kPa. Analysis Using the mm Hg to kPa units conversion factor,
P
(1000 mm Hg )
0.1333 kPa 1 mm Hg
133.3 kPa
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1-12
1-50 The pressure in a pressurized water tank is measured by a multi-fluid manometer. The gage pressure of air in the tank is to be determined. Assumptions The air pressure in the tank is uniform (i.e., its variation with elevation is negligible due to its low density), and thus we can determine the pressure at the air-water interface. Properties The densities of mercury, water, and oil are given to be 13,600, 1000, and 850 kg/m3, respectively. Analysis Starting with the pressure at point 1 at the air-water interface, and moving along the tube by adding (as we go down) or subtracting (as we go up) th e gh terms until we reach point 2, and setting the result equal to Patm since the tube is open to the atmosphere gives
P1
water gh1 oil gh2 mercury gh3
Patm
Solving for P1,
P1 Patm
water gh1 oil gh2 mercury gh3
or,
P1 Patm g(
mercury h3 water h1 oil h2 )
Noting that P1,gage = P1 - Patm and substituting,
P1,gage (9.81 m/s 2 )[(13,600 kg/m 3 )(0.46 m) (1000 kg/m 3 )(0.2 m) (850 kg/m 3 )(0.3 m)] 56.9 kPa 1N 1 kg m/s
2
1 kPa 1000 N/m 2
Discussion Note that jumping horizontally from one tube to the next and realizing that pressure remains the same in the same fluid simplifies the analysis greatly.
1-51 The barometric reading at a location is given in height of mercury column. The atmospheric pressure is to be determined. Properties The density of mercury is given to be 13,600 kg/m 3. Analysis The atmospheric pressure is determined directly from
Patm gh (13,600 kg/m 3 )(9.81 m/s 2 )(0.750 m) 100.1 kPa 1N 1 kg m/s
2
1 kPa 1000 N/m 2
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1-13
1-52 The gage pressure in a liquid at a certain depth is given. The gage pressure in the same liquid at a different depth is to be determined. Assumptions The variation of the density of the liquid with depth is negligible. Analysis The gage pressure at two different depths of a liquid can be expressed as
P1 gh1
and
P2
gh2
Taking their ratio,
P2 P1 gh2 gh1 h2 h1
h1 1 h2 2
84 kPa
Solving for P2 and substituting gives
P2 h2 P1 h1 9m (28 kPa) 3m
Discussion Note that the gage pressure in a given fluid is proportional to depth.
1-53 The absolute pressure in water at a specified depth is given. The local atmospheric pressure and the absolute pressure at the same depth in a different liquid are to be determined. Assumptions The liquid and water are incompressible. Properties The specific gravity of the fluid is given to be SG = 0.85. We take the density of water to be 1000 kg/m3. Then density of the liquid is obtained by multiplying its specific gravity by the density of water,
SG
H 2O
(0.85)(1000 kg/m 3 ) 850 kg/m3
Patm
Analysis (a) Knowing the absolute pressure, the atmospheric pressure can be determined from
Patm P gh 1 kPa 1000 N/m 2 (145 kPa) (1000 kg/m 3 )(9.81 m/s 2 )(5 m) 96.0 kPa
h P
(b) The absolute pressure at a depth of 5 m in the other liquid is
P Patm gh 1 kPa 1000 N/m 2 (96.0 kPa) (850 kg/m 3 )(9.81 m/s 2 )(5 m) 137.7 kPa
Discussion Note that at a given depth, the pressure in the lighter fluid is lower, as expected.
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1-14
1-54E It is to be shown that 1 kgf/cm2 = 14.223 psi . Analysis Noting that 1 kgf = 9.80665 N, 1 N = 0.22481 lbf, and 1 in = 2.54 cm, we have
1 kgf
and
9.80665 N (9.80665 N )
0.22481 lbf 1N
2.20463 lbf
1 kgf/cm 2
2.20463 lbf/cm 2
(2.20463 lbf/cm 2 )
2.54 cm 1 in
2
14.223 lbf/in 2
14.223 psi
1-55E The pressure in chamber 3 of the two-piston cylinder shown in the figure is to be determined. Analysis The area upon which pressure 1 acts is
A1 D12 4 (3 in) 2 4 7.069 in 2
F2
and the area upon which pressure 2 acts is
A2
2 D2 4
F3
(2 in) 4
2
3.142 in 2
The area upon which pressure 3 acts is given by
A3 A1 A2 7.069 3.142 3.927 in 2
The force produced by pressure 1 on the piston is then
F1 P1 A1 (150 psia ) 1 lbf/in 2 (7.069 in 2 ) 1060 lbf 1 psia
F1
while that produced by pressure 2 is
F1 P2 A2 (200 psia )(3.142 in 2 ) 628 lbf
According to the vertical force balance on the piston free body diagram
F3 F1 F2 1060 628 432 lbf
Pressure 3 is then
P3 F3 A3 432 lbf 3.927 in 2 110 psia
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1-15
1-56 The pressure in chamber 2 of the two-piston cylinder shown in the figure is to be determined. Analysis Summing the forces acting on the piston in the vertical direction gives
F2 P2 A2 P3 ( A1 F3 A2 ) F1 P1 A1
F2
which when solved for P2 gives
P2
P1
A1 A2
P3
A1 A2
1
D2 / 4
F3
since the areas of the piston faces are given by A the above equation becomes
P2 P1 D1 D2
2
P3 10 4
2
D1 D2
2
1
F1
10 4
2
(1000 kPa) 3625 kPa
(500 kPa)
1
1-57 The pressure in chamber 1 of the two-piston cylinder shown in the figure is to be determined. Analysis Summing the forces acting on the piston in the vertical direction gives
F2 P2 A2 P3 ( A1 F3 A2 ) F1 P1 A1
F2
which when solved for P1 gives
P1
P2
A2 A1
P3 1
A2 A1
D2 / 4
F3
since the areas of the piston faces are given by A the above equation becomes
P1 P2 D2 D1
2
P3 1 4 10
2
D2 D1
2
F1
4 10
2
(2000 kPa) 908 kPa
(700 kPa) 1
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1-16
1-58 The mass of a woman is given. The minimum imprint area per shoe needed to enable her to walk on the snow without sinking is to be determined. Assumptions 1 The weight of the person is distributed uniformly on the imprint area of the shoes. 2 One foot carries the entire weight of a person during walking, and the shoe is sized for walking conditions (rather than standing). 3 The weight of the shoes is negligible. Analysis The mass of the woman is given to be 70 kg. For a pressure of 0.5 kPa on the snow, the imprint area of one shoe must be
A W P mg P 1 kPa 1000 N/m 2 1.37 m 2
(70 kg)(9.81 m/s 2 ) 1N 0.5 kPa 1 kg m/s 2
Discussion This is a very large area for a shoe, and such shoes would be impractical to use. Therefore, some sinking of the snow should be allowed to have shoes of reasonable size.
1-59 The vacuum pressure reading of a tank is given. The absolute pressure in the tank is to be determined. Properties The density of mercury is given to be = 13,590 kg/m3.
Analysis The atmospheric (or barometric) pressure can be expressed as
Patm gh (13,590 kg/m )(9.807 m/s )(0.750 m) 100.0 kPa
3 2
1N 1 kg m/s 2
1 kPa 1000 N/m 2
Pabs
15 kPa kPa
Then the absolute pressure in the tank becomes
Pabs Patm Pvac 100.0 15 85.0 kPa
Patm = 750 mmHg
1-60E A pressure gage connected to a tank reads 50 psi. The absolute pressure in the tank is to be determined. Properties The density of mercury is given to be = 848.4 lbm/ft3.
Analysis The atmospheric (or barometric) pressure can be expressed as
Patm gh (848.4 lbm/ft 3 )(32.2 ft/s 2 )(29.1/12 ft) 14.29 psia 1 lbf 32.2 lbm ft/s
2
1 ft 2 144 in
2
Pabs
50 psi
Then the absolute pressure in the tank is
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1-17
Pabs
Pgage
Patm
50 14.29 64.3 psia
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1-18
1-61 A pressure gage connected to a tank reads 500 kPa. The absolute pressure in the tank is to be determined. Analysis The absolute pressure in the tank is determined from
Pabs Pgage Patm 500 94 594 kPa
Pabs
500 kPa
Patm = 94 kPa
1-62 A mountain hiker records the barometric reading before and after a hiking trip. The vertical distance climbed is to be determined. 780 mbar Assumptions The variation of air density and the gravitational acceleration with altitude is negligible. Properties The density of air is given to be = 1.20 kg/m3. h=?
Analysis Taking an air column between the top and the bottom of the mountain and writing a force balance per unit base area, we obtain
Wair / A ( gh) air (1.20 kg/m 3 )(9.81 m/s 2 )(h) 1N 1 kg m/s
2
930 mbar
Pbottom Pbottom Ptop Ptop
1 bar 100,000 N/m 2
(0.930 0.780) bar
It yields h = 1274 m which is also the distance climbed.
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1-19
1-63 A barometer is used to measure the height of a building by recording reading at the bottom and at the top of the building. The height of the building is to be determined. Assumptions The variation of air density with altitude is negligible. Properties The density of air is given to be density of mercury is 13,600 kg/m3. = 1.18 kg/m3. The 730 mmHg
Analysis Atmospheric pressures at the top and at the bottom of the building are
Ptop ( g h) top (13,600 kg/m 3 )(9.807 m/s 2 )(0.730 m) 97.36 kPa Pbottom ( g h) bottom (13,600 kg/m 3 )(9.807 m/s 2 )(0.755 m) 100.70 kPa 1N 1 kg m/s
2
h
1 kPa 1000 N/m 2
755 mmHg
1N 1kg m/s
2
1 kPa 1000 N/m 2
Taking an air column between the top and the bottom of the building and writing a force balance per unit base area, we obtain
Wair / A ( gh) air (1.18 kg/m 3 )(9.807 m/s 2 )(h) 1N 1 kg m/s
2
Pbottom Pbottom
Ptop Ptop
1 kPa 1000 N/m 2
(100.70 97.36) kPa
It yields
h = 288.6 m
which is also the height of the building.
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1-20
1-64 EES Problem 1-63 is reconsidered. The entire EES solution is to be printed out, including the numerical results with proper units. Analysis The problem is solved using EES, and the solution is given below. P_bottom=755 [mmHg] P_top=730 [mmHg] g=9.807 [m/s^2] "local acceleration of gravity at sea level" rho=1.18 [kg/m^3] DELTAP_abs=(P_bottom-P_top)*CONVERT('mmHg','kPa')"[kPa]" "Delta P reading from the barometers, converted from mmHg to kPa." DELTAP_h =rho*g*h/1000 "[kPa]" "Equ. 1-16. Delta P due to the air fluid column height, h, between the top and bottom of the building." "Instead of dividing by 1000 Pa/kPa we could have multiplied rho*g*h by the EES function, CONVERT('Pa','kPa')" DELTAP_abs=DELTAP_h SOLUTION Variables in Main DELTAP_abs=3.333 [kPa] DELTAP_h=3.333 [kPa] g=9.807 [m/s^2] h=288 [m] P_bottom=755 [mmHg] P_top=730 [mmHg] rho=1.18 [kg/m^3]
1-65 A diver is moving at a specified depth from the water surface. The pressure exerted on the surface of the diver by water is to be determined. Assumptions The variation of the density of water with depth is negligible. Properties The specific gravity of seawater is given to be SG = 1.03. We take the density of water to be 1000 kg/m3. Analysis The density of the seawater is obtained by multiplying its specific gravity by the density of water which is taken to be 1000 kg/m3:
SG
H 2O
Patm Sea h P
(1.03)(1000 kg/m )
3
1030 kg/m
3
The pressure exerted on a diver at 30 m below the free surface of the sea is the absolute pressure at that location:
P Patm gh 1 kPa 1000 N/m 2 (101 kPa) (1030 kg/m 3 )(9.807 m/s 2 )(30 m) 404.0 kPa
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1-21
1-66 A gas contained in a vertical piston-cylinder device is pressurized by a spring and by the weight of the piston. The pressure of the gas is to be determined. Analysis Drawing the free body diagram of the piston and balancing the vertical forces yield
PA Patm A W Fspring
Fspring Patm
Thus,
P Patm mg Fspring A (4 kg)(9.81 m/s 2 ) 60 N 35 10
4
P
1 kPa 1000 N/m 2
(95 kPa)
123.4 kPa
m2
W = mg
1-67 EES Problem 1-66 is reconsidered. The effect of the spring force in the range of 0 to 500 N on the pressure inside the cylinder is to be investigated. The pressure against the spring force is to be plotted, and results are to be discussed. Analysis The problem is solved using EES, and the solution is given below. g=9.807 [m/s^2] P_atm= 95 [kPa] m_piston=4 [kg] {F_spring=60 [N]} A=35*CONVERT('cm^2','m^2')"[m^2]" W_piston=m_piston*g"[N]" F_atm=P_atm*A*CONVERT('kPa','N/m^2')"[N]" "From the free body diagram of the piston, the balancing vertical forces yield:" F_gas= F_atm+F_spring+W_piston"[N]" P_gas=F_gas/A*CONVERT('N/m^2','kPa')"[kPa]"
Fspring [N] 0 55.56 111.1 166.7 222.2 277.8 333.3 388.9 444.4 500
Pgas [kPa] 106.2 122.1 138 153.8 169.7 185.6 201.4 217.3 233.2 249.1
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1-22
1-68 [Also solved by EES on enclosed CD] Both a gage and a manometer are attached to a gas to measure its pressure. For a specified reading of gage pressure, the difference between the fluid levels of the two arms of the manometer is to be determined for mercury and water. Properties The densities of water and mercury are given to be 3 3 water = 1000 kg/m and be Hg = 13,600 kg/m . Analysis The gage pressure is related to the vertical distance h between the two fluid levels by
Pgage gh h Pgage g
(a) For mercury,
h Pgage
Hg g
1 kN/m 2 (13,600 kg/m 3 )(9.81 m/s 2 ) 1 kPa 80 kPa
1000 kg/m s 2 1 kN
0.60 m
(b) For water,
h Pgage
H 2O g
1 kN/m 2 (1000 kg/m 3 )(9.81 m/s 2 ) 1 kPa 80 kPa
1000 kg/m s 2 1 kN
8.16 m
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1-23
1-69 EES Problem 1-68 is reconsidered. The effect of the manometer fluid density in the range of 800 to 13,000 kg/m3 on the differential fluid height of the manometer is to be investigated. Differential fluid height against the density is to be plotted, and the results are to be discussed. Analysis The problem is solved using EES, and the solution is given below. Function fluid_density(Fluid$) If fluid$='Mercury' then fluid_density=13600 else fluid_density=1000 end {Input from the diagram window. If the diagram window is hidden, then all of the input must come from the equations window. Also note that brackets can also denote comments - but these comments do not appear in the formatted equations window.} {Fluid$='Mercury' P_atm = 101.325 "kpa" DELTAP=80 "kPa Note how DELTAP is displayed on the Formatted Equations Window."} g=9.807 "m/s2, local acceleration of gravity at sea level" rho=Fluid_density(Fluid$) "Get the fluid density, either Hg or H2O, from the function" "To plot fluid height against density place {} around the above equation. Then set up the parametric table and solve." DELTAP = RHO*g*h/1000 "Instead of dividing by 1000 Pa/kPa we could have multiplied by the EES function, CONVERT('Pa','kPa')" h_mm=h*convert('m','mm') "The fluid height in mm is found using the built-in CONVERT function." P_abs= P_atm + DELTAP
hmm [mm] 10197 3784 2323 1676 1311 1076 913.1 792.8 700.5 627.5
[kg/m3] 800 2156 3511
Manometer Fluid Height vs Manometer Fluid Density
11000
8800
h mm [mm]
4867 6222 7578 8933 10289 11644 13000
6600
4400
2200
0 0
2000
4000
6000
8000
10000 12000 14000
[kg/m^3]
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1-24
1-70 The air pressure in a tank is measured by an oil manometer. For a given oil-level difference between the two columns, the absolute pressure in the tank is to be determined. Properties The density of oil is given to be = 850 kg/m3.
Analysis The absolute pressure in the tank is determined from
P Patm gh 1kPa 1000 N/m 2 (98 kPa) (850 kg/m 3 )(9.81m/s 2 )(0.60 m) 103 kPa
AIR
0.60 m
Patm = 98 kPa
1-71 The air pressure in a duct is measured by a mercury manometer. For a given mercury-level difference between the two columns, the absolute pressure in the duct is to be determined. Properties The density of mercury is given to be = 13,600 kg/m3.
Analysis (a) The pressure in the duct is above atmospheric pressure since the fluid column on the duct side is at a lower level. (b) The absolute pressure in the duct is determined from
P Patm gh 1N 1 kg m/s
2
(100 kPa) (13,600 kg/m 3 )(9.81 m/s 2 )(0.015 m) 102 kPa
1 kPa 1000 N/m 2
1-72 The air pressure in a duct is measured by a mercury manometer. For a given mercury-level difference between the two columns, the absolute pressure in the duct is to be determined. Properties The density of mercury is given to be = 13,600 kg/m3. AIR P
1N 1 kg m/s
2
Analysis (a) The pressure in the duct is above atmospheric pressure since the fluid column on the duct side is at a lower level. (b) The absolute pressure in the duct is determined from
P Patm gh 1 kPa 1000 N/m 2 (100 kPa) (13,600 kg/m 3 )(9.81 m/s 2 )(0.045 m) 106 kPa
45 mm
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1-25
1-73E The systolic and diastolic pressures of a healthy person are given in mmHg. These pressures are to be expressed in kPa, psi, and meter water column. Assumptions Both mercury and water are incompressible substances. Properties We take the densities of water and mercury to be 1000 kg/m 3 and 13,600 kg/m3, respectively. Analysis Using the relation P
Phigh Plow ghhigh ghlow
gh for gage pressure, the high and low pressures are expressed as
1N 1 kg m/s 2 1N 1 kg m/s
2
(13,600 kg/m 3 )(9.81 m/s 2 )(0.12 m) (13,600 kg/m 3 )(9.81 m/s 2 )(0.08 m)
1 kPa 1000 N/m 2 1 kPa 1000 N/m 2
16.0 kPa
10.7 kPa
Noting that 1 psi = 6.895 kPa,
Phigh
(16.0 Pa)
1 psi 6.895 kPa
2.32 psi
and
Plow
(10.7 Pa)
1 psi 6.895 kPa
1.55 psi
For a given pressure, the relation P gh can be expressed for mercury and water as P water gh water and P ghmercury . Setting these two relations equal to each mercury other and solving for water height gives
P
water
ghwater
mercury ghmercury
hwater
mercury water
hmercury
h
Therefore,
hwater, high hwater, low
mercury water mercury water
hmercury, high hmercury, low
13,600 kg/m 3 1000 kg/m 3 13,600 kg/m 3 1000 kg/m 3
(0.12 m) 1.63 m
(0.08 m) 1.09 m
Discussion Note that measuring blood pressure with a "water" monometer would involve differential fluid heights higher than the person, and thus it is impractical. This problem shows why mercury is a suitable fluid for blood pressure measurement devices.
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1-26
1-74 A vertical tube open to the atmosphere is connected to the vein in the arm of a person. The height that the blood will rise in the tube is to be determined. Assumptions 1 The density of blood is constant. 2 The gage pressure of blood is 120 mmHg. Properties The density of blood is given to be = 1050 kg/m3.
Analysis For a given gage pressure, the relation P for mercury and blood as P blood gh blood and P Setting these two relations equal to each other we get
P
blood ghblood mercury ghmercury
gh can be expressed mercury ghmercury .
Blood h
Solving for blood height and substituting gives
hblood
mercury blood
hmercury
13,600 kg/m 3 1050 kg/m 3
(0.12 m) 1.55 m
Discussion Note that the blood can rise about one and a half meters in a tube connected to the vein. This explains why IV tubes must be placed high to force a fluid into the vein of a patient.
1-75 A man is standing in water vertically while being completely submerged. The difference between the pressures acting on the head and on the toes is to be determined. Assumptions Water is an incompressible substance, and thus the density does not change with depth. Properties We take the density of water to be =1000 kg/m3.
hhead
Analysis The pressures at the head and toes of the person can be expressed as
Phead Patm ghhead
and
Ptoe
Patm
ghtoe
where h is the vertical distance of the location in water from the free surface. The pressure difference between the toes and the head is determined by subtracting the first relation above from the second,
Ptoe Phead ghtoe ghhead g (htoe hhead )
htoe
Substituting,
Ptoe Phead (1000 kg/m 3 )(9.81 m/s 2 )(1.80 m - 0) 1N 1kg m/s
2
1kPa 1000 N/m 2
17.7 kPa
Discussion This problem can also be solved by noting that the atmospheric pressure (1 atm = 101.325 kPa) is equivalent to 10.3-m of water height, and finding the pressure that corresponds to a water height of 1.8 m.
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1-27
1-76 Water is poured into the U-tube from one arm and oil from the other arm. The water column height in one arm and the ratio of the heights of the two fluids in the other arm are given. The height of each fluid in that arm is to be determined. Assumptions Both water and oil are incompressible substances. Properties The density of oil is given to be the density of water to be =1000 kg/m3. = 790 kg/m3. We take ha hw1 hw2 Water oil
Analysis The height of water column in the left arm of the monometer is given to be hw1 = 0.70 m. We let the height of water and oil in the right arm to be hw2 and ha, respectively. Then, ha = 4hw2. Noting that both arms are open to the atmosphere, the pressure at the bottom of the U-tube can be expressed as
Pbottom Patm
w gh w1
and
Pbottom
Patm
w ghw2
a gha
Setting them equal to each other and simplifying,
w gh w1 w gh w2 a gha w hw1 w hw2 a ha
hw1
hw2
(
a
/
w )ha
Noting that ha = 4hw2, the water and oil column heights in the second arm are determined to be
0.7 m 0.7 m hw2 (790/1000) 4hw2 hw2 ha 0.168 m 0.673 m
0.168 m (790/1000) ha
Discussion Note that the fluid height in the arm that contains oil is higher. This is expected since oil is lighter than water.
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1-28
1-77 Fresh and seawater flowing in parallel horizontal pipelines are connected to each other by a double U-tube manometer. The pressure difference between the two pipelines is to be determined. Assumptions 1 All the liquids are incompressible. 2 The effect of air column on pressure is negligible. Properties The densities of seawater and mercury are given to be sea = 1035 kg/m3 and Hg = 13,600 kg/m3. We take the density of water to be w =1000 kg/m3. Analysis Starting with the pressure in the fresh water pipe (point 1) and moving along the tube by adding (as we go down) or subtracting (as we go up) the gh terms until we reach the sea water pipe (point 2), and setting the result equal to P2 gives
P1
w ghw Hg ghHg air ghair sea ghsea
Air
hsea
Fresh Water hw
hair hHg
Sea Water
P2
Mercury
w hw sea hsea )
Rearranging and neglecting the effect of air column on pressure,
P1 P2
w ghw Hg ghHg sea ghsea
g(
Hg hHg
Substituting,
P1 P2 (9.81 m/s 2 )[(13600 kg/m 3 )(0.1 m) (1000 kg/m 3 )(0.6 m) (1035 kg/m 3 )(0.4 m)] 3.39 kN/m 2 3.39 kPa 1 kN 1000 kg m/s 2
Therefore, the pressure in the fresh water pipe is 3.39 kPa higher than the pressure in the sea water pipe. Discussion A 0.70-m high air column with a density of 1.2 kg/m 3 corresponds to a pressure difference of 0.008 kPa. Therefore, its effect on the pressure difference between the two pipes is negligible.
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1-29
1-78 Fresh and seawater flowing in parallel horizontal pipelines are connected to each other by a double U-tube manometer. The pressure difference between the two pipelines is to be determined. Assumptions All the liquids are incompressible. Properties The densities of seawater and mercury are given to be sea = 1035 kg/m3 and Hg = 13,600 kg/m3. We take the density of water to be w =1000 kg/m3. The specific gravity of oil is given to be 0.72, and thus its density is 720 kg/m3. Analysis Starting with the pressure in the fresh water pipe (point 1) and moving along the tube by adding (as we go down) or subtracting (as we go up) the gh terms until we reach the sea water pipe (point 2), and setting the result equal to P2 gives
P1
w ghw Hg ghHg oil ghoil sea ghsea
Oil
hsea
Fresh Water hw
hoil
Sea Water
hHg
P2
Mercury
Rearranging,
P1 P2 g(
Substituting,
P1 P2
w ghw Hg hHg
Hg ghHg oil hoil
oil ghoil w hw
sea ghsea sea hsea )
(9.81 m/s 2 )[(13600 kg/m 3 )(0.1 m) (720 kg/m 3 )(0.7 m) (1000 kg/m 3 )(0.6 m) (1035 kg/m 3 )(0.4 m)] 8.34 kN/m 2 8.34 kPa 1 kN 1000 kg m/s 2
Therefore, the pressure in the fresh water pipe is 8.34 kPa higher than the pressure in the sea water pipe.
1-79 The pressure indicated by a manometer is to be determined. Properties The specific weights of fluid A and fluid B are given to be 10 kN/m3 and 8 kN/m3, respectively. Analysis The absolute pressure P1 is determined from
P1 Patm Patm ( gh) A
AhA
( gh) B
B hB
= hB hA =
0.1333 kPa (758 mm Hg) 1 mm Hg (10 kN/m 3 )(0.05 m) (8 kN/m 3 )(0.15 m) 102.7 kPa
Note that 1 kPa = 1 kN/m2.
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1-30
1-80 The pressure indicated by a manometer is to be determined. Properties The specific weights of fluid A and fluid B are given to be 100 kN/m 3 and 8 kN/m3, respectively. Analysis The absolute pressure P1 is determined from
P1 Patm Patm ( gh) A
AhA
( gh) B
B hB
90 kPa (100 kN/m 3 )(0.05 m) (8 kN/m 3 )(0.15 m) 96.2 kPa
= hB hA =
Note that 1 kPa = 1 kN/m2.
100 kN/m3
1-81 The pressure indicated by a manometer is to be determined. Properties The specific weights of fluid A and fluid B are given to be 10 kN/m 3 and 20 kN/m3, respectively. Analysis The absolute pressure P1 is determined from
P1 Patm Patm ( gh) A
AhA
( gh) B
B hB
= hB hA =
3
0.1333 kPa (745 mm Hg) 1 mm Hg (10 kN/m )(0.05 m) (20 kN/m )(0.15 m) 102.8 kPa
3
20 kN/m3
Note that 1 kPa = 1 kN/m2.
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1-31
1-82 The gage pressure of air in a pressurized water tank is measured simultaneously by both a pressure gage and a manometer. The differential height h of the mercury column is to be determined. Assumptions The air pressure in the tank is uniform (i.e., its variation with elevation is negligible due to its low density), and thus the pressure at the air-water interface is the same as the indicated gage pressure. Properties We take the density of water to be are given to be 0.72 and 13.6, respectively.
w =1000
kg/m3. The specific gravities of oil and mercury
Analysis Starting with the pressure of air in the tank (point 1), and moving along the tube by adding (as we go down) or subtracting (as we go u p) the gh terms until we reach the free surface of oil where the oil tube is exposed to the and atmosphere, setting the result equal to Patm gives
P1
w ghw Hg ghHg oil ghoil
Patm
Rearranging,
P1 Patm
oil ghoil Hg ghHg w ghw
or,
P1,gage
wg
SG oil hoil
SG Hg hHg
hw
Substituting,
80 kPa (1000 kg/m 3 )(9.81 m/s 2 ) 1000 kg m/s 2 1 kPa. m 2 0.72 (0.75 m) 13.6 hHg 0.3 m
Solving for hHg gives hHg = 0.582 m. Therefore, the differential height of the mercury column must be 58.2 cm. Discussion Double instrumentation like this allows one to verify the measurement of one of the instruments by the measurement of another instrument.
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1-32
1-83 The gage pressure of air in a pressurized water tank is measured simultaneously by both a pressure gage and a manometer. The differential height h of the mercury column is to be determined. Assumptions The air pressure in the tank is uniform (i.e., its variation with elevation is negligible due to its low density), and thus the pressure at the air-water interface is the same as the indicated gage pressure. Properties We take the density of water to be are given to be 0.72 and 13.6, respectively.
w =1000
kg/m3. The specific gravities of oil and mercury
Analysis Starting with the pressure of air in the tank (point 1), and moving along the tube by adding (as we go down) or subtracting (as we go up) the gh terms until we reach the free surface of oil where the oil tube is exposed to the atmosphere, and setting the result equal to Patm gives
P1
w ghw Hg ghHg oil ghoil
Patm
Rearranging,
P1 Patm
oil ghoil Hg ghHg w ghw
40 kPa AIR hoil
SG oil hoil SG
Hg hHg
or, Substituting,
P1,gage
wg
hw
Water hw hHg
40 kPa (1000 kg/m 3 )(9.81 m/s 2 )
1000 kg m/s 2 1 kPa. m 2
0.72 (0.75 m) 13.6 hHg
0.3 m
Solving for hHg gives hHg = 0.282 m. Therefore, the differential height of the mercury column must be 28.2 cm. Discussion Double instrumentation like this allows one to verify the measurement of one of the instruments by the measurement of another instrument.
1-84 The top part of a water tank is divided into two compartments, and a fluid with an unknown density is poured into one side. The levels of the water and the liquid are measured. The density of the fluid is to be determined. Assumptions 1 Both water and the added liquid are incompressible substances. 2 The added liquid does not mix with water. Properties We take the density of water to be =1000 kg/m3. hf hw Fluid Water
Analysis Both fluids are open to the atmosphere. Noting that the pressure of both water and the added fluid is the same at the contact surface, the pressure at this surface can be expressed as
Pcontact Patm
f
ghf
f
Patm
w gh w
Simplifying and solving for
f
gives
hw hf
w
ghf
w ghw
f
45 cm (1000 kg/m 3 ) 80 cm
562.5 kg/m3
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1-33
Discussion Note that the added fluid is lighter than water as expected (a heavier fluid would sink in water).
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1-34
1-85 The fluid levels in a multi-fluid U-tube manometer change as a result of a pressure drop in the trapped air space. For a given pressure drop and brine level change, the area ratio is to be determined. Assumptions 1 All the liquids are incompressible. 2 Pressure in the brine pipe remains constant. 3 The variation of pressure in the trapped air space is negligible. Properties The specific gravities are given to be 13.56 for mercury and 1.1 for brine. We take the standard density of water to be w =1000 kg/m3. Analysis It is clear from the problem statement and the figure that the brine pressure is much higher than the air pressure, and when the air pressure drops by 0.7 kPa, the pressure difference between the brine and the air space increases also by the same amount. Area, A1
A Air B Brine pipe SG=1.1 Mercury SG=13.56 hb = 5 mm
Water
Area, A2
Starting with the air pressure (point A) and moving along the tube by adding (as we go down) or subtracting (as we go up) the gh terms until we reach the brine pipe (point B), and setting the result equal to PB before and after the pressure change of air give Before: After:
PA1
w ghw Hg ghHg, 1 br ghbr,1
PB
PA2
w ghw
Hg ghHg, 2
br ghbr,2
PB
Subtracting,
PA2 PA1
Hg g
hHg
br g
hbr
0
PA1
PA2 g w
SG Hg hHg
SG br hbr
0
(1)
where
hHg and
h br are the changes in the differential mercury and brine column heights, respectively,
due to the drop in air pressure. Both of these are positive quantities since as the mercury-brine interface drops, the differential fluid heights for both mercury and brine increase. Noting also that the volume of mercury is constant, we have A1 hHg,left A2 hHg,right and
PA2 PA1 0.7 kPa 700 N/m 2 700 kg/m s 2
hbr
0.005 m
hHg
hHg,right
hHg,left
hbr
hbr A2 /A1
hbr (1 A2 /A1 )
Substituting,
700 kg/m s 2 (1000 kg/m 3 )(9.81 m/s 2 )
It gives A2/A1 = 0.134
[13.56 0.005(1 A2 /A1 ) (1.1 0.005)] m
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1-35
1-86 A multi-fluid container is connected to a U-tube. For the given specific gravities and fluid column heights, the gage pressure at A and the height of a mercury column that would create the same pressure at A are to be determined. Assumptions 1 All the liquids are incompressible. 2 The multi-fluid container is open to the atmosphere. Properties The specific gravities are given to be 1.26 for glycerin and 0.90 for oil. We take the standard density of water to be w =1000 kg/m3, and the specific gravity of mercury to be 13.6. Analysis Starting with the atmospheric pressure on the top surface of the container and moving along the tube by adding (as we go down) or subtracting (as we go up) the gh terms until we reach point A, and setting the result equal to PA give
Patm
oil ghoil w ghw gly ghgly
70 cm
Oil SG=0.90
A
30 cm
Water Glycerin SG=1.26
90 cm
20 cm
15 cm
PA
Rearranging and using the definition of specific gravity,
PA Patm SG oil
w ghoil
SG w
w ghw
SG gly
w ghgly
or
PA,gage g
w (SG oil hoil
SG w hw SG gly hgly )
Substituting,
PA,gage (9.81 m/s 2 )(1000 kg/m 3 )[0.90(0.70 m) 1(0.3 m) 1.26(0.70 m)] 0.471 kN/m 2 0.471kPa 1 kN 1000 kg m/s 2
The equivalent mercury column height is
hHg PA,gage
Hg g
1000 kg m/s 2 1 kN (13,600 kg/m 3 )(1000 kg/m 3 )(9.81 m/s 2 ) 0.471 kN/m 2
0.00353 m 0.353 cm
Discussion Note that the high density of mercury makes it a very suitable fluid for measuring high pressures in manometers.
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Solving Engineering Problems and EES
1-87C Despite the convenience and capability the engineering software packages offer, they are still just tools, and they will not replace the traditional engineering courses. They will simply cause a shift in emphasis in the course material from mathematics to physics. They are of great value in engineering practice, however, as engineers today rely on software packages for solving large and complex problems in a short time, and perform optimization studies efficiently.
1-88 EES Determine a positive real root of the following equation using EES: 2x3 10x0.5 3x = -3 Solution by EES Software (Copy the following line and paste on a blank EES screen to verify solution): 2*x^3-10*x^0.5-3*x = -3 Answer: x = 2.063 (using an initial guess of x=2)
1-89 EES Solve the following system of 2 equations with 2 unknowns using EES: x3 y2 = 7.75 3xy + y = 3.5 Solution by EES Software (Copy the following lines and paste on a blank EES screen to verify solution): x^3-y^2=7.75 3*x*y+y=3.5 Answer x=2 y=0.5
1-90 EES Solve the following system of 3 equations with 3 unknowns using EES: 2x y + z = 5 3x2 + 2y = z + 2 xy + 2z = 8 Solution by EES Software (Copy the following lines and paste on a blank EES screen to verify solution): 2*x-y+z=5 3*x^2+2*y=z+2 x*y+2*z=8 Answer x=1.141, y=0.8159, z=3.535
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1-91 EES Solve the following system of 3 equations with 3 unknowns using EES: x2y z = 1 x 3y
0.5
+ xz = - 2
x+yz=2 Solution by EES Software (Copy the following lines and paste on a blank EES screen to verify solution): x^2*y-z=1 x-3*y^0.5+x*z=-2 x+y-z=2 Answer x=1, y=1, z=0
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1-92E EES Specific heat of water is to be expressed at various units using unit conversion capability of EES. Analysis The problem is solved using EES, and the solution is given below.
EQUATION WINDOW "GIVEN" C_p=4.18 [kJ/kg-C] "ANALYSIS" C_p_1=C_p*Convert(kJ/kg-C, kJ/kg-K) C_p_2=C_p*Convert(kJ/kg-C, Btu/lbm-F) C_p_3=C_p*Convert(kJ/kg-C, Btu/lbm-R) C_p_4=C_p*Convert(kJ/kg-C, kCal/kg-C) FORMATTED EQUATIONS WINDOW
GIVEN C p = 4.18 ANALYSIS C p,1 = Cp 1 kJ/kgK kJ/kgC Btu/lbmF kJ/kgC Btu/lbmR kJ/kgC kCal/kgC kJ/kgC [kJ/kg-C]
C p,2 =
Cp
0.238846
C p,3 =
Cp
0.238846
C p,4 =
Cp
0.238846
SOLUTION WINDOW C_p=4.18 [kJ/kg-C] C_p_1=4.18 [kJ/kg-K] C_p_2=0.9984 [Btu/lbm-F] C_p_3=0.9984 [Btu/lbm-R] C_p_4=0.9984 [kCal/kg-C]
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Review Problems
1-93 The weight of a lunar exploration module on the moon is to be determined. Analysis Applying Newton's second law, the weight of the module on the moon can be determined from
W moon mg moon Wearth g moon g earth 4000 N 9.8 m/s
2
(1.64 m/s 2 )
669 N
1-94 The deflection of the spring of the two-piston cylinder with a spring shown in the figure is to be determined. Analysis Summing the forces acting on the piston in the vertical direction gives
Fs kx P2 A2 F2 F3 A2 ) F1 P1 A1
F2
P3 ( A1
which when solved for the deflection of the spring and substituting A D 2 / 4 gives
x 4k P1 D12
2 P2 D 2
Fs
F3
P3 ( D12
2 D2 )
5000 0.08 2 10,000 0.03 2 1000(0.08 2 4 800 0.0172 m 1.72 cm
0.03 2 )
F1
We expressed the spring constant k in kN/m, the pressures in kPa (i.e., kN/m 2) and the diameters in m units.
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1-95 The pressure in chamber 1 of the two-piston cylinder with a spring shown in the figure is to be determined. Analysis Summing the forces acting on the piston in the vertical direction gives
Fs F1 F2 P2 A2 F3 P3 ( A1 A2 )
D2 / 4
F2
kx P1 A1
which when solved for the P3 and substituting A gives
P1 P2 A2 A1 D2 D1 P3 1
2
Fs
F3
A2 A1
kx A1 D2 D1
2
P2
P3 1 3 7
2
4kx D12 3 7
2
F1
4(1200 kN/m)(0.05 m) (0.07 m) 2
(8000 kPa) 13,880 kPa
(300 kPa) 1
13.9 MPa
1-96E The pressure in chamber 2 of the two-piston cylinder with a spring shown in the figure is to be determined. Analysis The areas upon which pressures act are
A1 D12 4
2 D2 4
(5 in) 2 4
19.63 in 2
F2
A2
(2 in) 2 4
3.142 in 2
Fs
F3
A3
A1
A2
19.63 3.142 16.49 in 2
The forces generated by pressure 1 and 3 are
F1 P1 A1 (100 psia ) 1 lbf/in 2 (19.63 in 2 ) 1963 lbf 1 psia
F1
F3 P2 A2 (20 psia )(16.49 in )
2
330 lbf
The force exerted by the spring is
Fs kx (200 lbf/in )(2 in ) 400 lbf
Summing the vertical forces acting on the piston gives
F2 F1 F3 Fs 1963 330 400 1233 lbf
The pressure at 2 is then
P2 F2 A2 1233 lbf 3.142 in 2 392 psia
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1-97 An airplane is flying over a city. The local atmospheric pressure in that city is to be determined. Assumptions The gravitational acceleration does not change with altitude. Properties The densities of air and mercury are given to be 1.15 kg/m 3 and 13,600 kg/m3. Analysis The local atmospheric pressure is determined from
Patm Pplane gh 1 kN 1000 kg m/s 2 91.84 kN/m 2 91.8 kPa
58 kPa (1.15 kg/m 3 )(9.81m/s 2 )(3000 m)
The atmospheric pressure may be expressed in mmHg as
hHg Patm g 1000 Pa (13,600 kg/m )(9.81m/s ) 1 kPa
3 2
91.8 kPa
1000 mm 1m
688 mmHg
1-98 The gravitational acceleration changes with altitude. Accounting for this variation, the weights of a body at different locations are to be determined. Analysis The weight of an 80-kg man at various locations is obtained by substituting the altitude z (values in m) into the relation
W mg (80kg)(9.807 3.32 10
6
z m/s 2 )
1N 1kg m/s 2
Sea level: Denver: Mt. Ev.:
(z = 0 m): W = 80 (9.807-3.32x10-6 0) = 80 9.807 = 784.6 N (z = 1610 m): W = 80 (9.807-3.32x10-6 1610) = 80 9.802 = 784.2 N (z = 8848 m): W = 80 (9.807-3.32x10-6 8848) = 80 9.778 = 782.2 N
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1-99 A man is considering buying a 12-oz steak for $3.15, or a 320-g steak for $2.80. The steak that is a better buy is to be determined. Assumptions The steaks are of identical quality. Analysis To make a comparison possible, we need to express the cost of each steak on a common basis. Let us choose 1 kg as the basis for comparison. Using proper conversion factors, the unit cost of each steak is determined to be 12 ounce steak:
Unit Cost =
320 gram steak:
$3.15 12 oz
16 oz 1 lbm
1 lbm 0.45359 kg
$9.26/kg
Unit Cost =
$2.80 320 g
1000 g 1 kg
$8.75/kg
Therefore, the steak at the international market is a better buy.
1-100E The thrust developed by the jet engine of a Boeing 777 is given to be 85,000 pounds. This thrust is to be expressed in N and kgf. Analysis Noting that 1 lbf = 4.448 N and 1 kgf = 9.81 N, the thrust developed can be expressed in two other units as Thrust in N:
Thrust (85,000 lbf )
4.448 N 1 lbf
1 kgf 9.81 N
3.78 10 5 N
3.85 10 4 kgf
Thrust in kgf:
Thrust (37.8 10 5 N)
1-101E The efficiency of a refrigerator increases by 3% per C rise in the minimum temperature. This increase is to be expressed per F, K, and R rise in the minimum temperature. Analysis The magnitudes of 1 K and 1 C are identical, so are the magnitudes of 1 R and 1 F. Also, a change of 1 K or 1 C in temperature corresponds to a change of 1.8 R or 1.8 F. Therefore, the increase in efficiency is (a) 3% for each K rise in temperature, and (b), (c) 3/1.8 = 1.67% for each R or F rise in temperature.
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1-102E The boiling temperature of water decreases by 3 C for each 1000 m rise in altitude. This decrease in temperature is to be expressed in F, K, and R. Analysis The magnitudes of 1 K and 1 C are identical, so are the magnitudes of 1 R and 1 F. Also, a change of 1 K or 1 C in temperature corresponds to a change of 1.8 R or 1.8 F. Therefore, the decrease in the boiling temperature is (a) 3 K for each 1000 m rise in altitude, and (b), (c) 3 1.8 = 5.4 F = 5.4 R for each 1000 m rise in altitude.
1-103E Hyperthermia of 5 C is considered fatal. This fatal level temperature change of body temperature is to be expressed in F, K, and R. Analysis The magnitudes of 1 K and 1 C are identical, so are the magnitudes of 1 R and 1 F. Also, a change of 1 K or 1 C in temperature corresponds to a change of 1.8 R or 1.8 F. Therefore, the fatal level of hypothermia is (a) 5 K (b) 5 1.8 = 9 F (c) 5 1.8 = 9 R
1-104E A house is losing heat at a rate of 4500 kJ/h per C temperature difference between the indoor and the outdoor temperatures. The rate of heat loss is to be expressed per F, K, and R of temperature difference between the indoor and the outdoor temperatures. Analysis The magnitudes of 1 K and 1 C are identical, so are the magnitudes of 1 R and 1 F. Also, a change of 1 K or 1 C in temperature corresponds to a change of 1.8 R or 1.8 F. Therefore, the rate of heat loss from the house is (a) 4500 kJ/h per K difference in temperature, and (b), (c) 4500/1.8 = 2500 kJ/h per R or F rise in temperature.
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1-105 The average temperature of the atmosphere is expressed as Tatm = 288.15 6.5z where z is altitude in km. The temperature outside an airplane cruising at 12,000 m is to be determined. Analysis Using the relation given, the average temperature of the atmosphere at an altitude of 12,000 m is determined to be Tatm = 288.15 - 6.5z = 288.15 - 6.5 12 = 210.15 K = - 63 C Discussion This is the "average" temperature. The actual temperature at different times can be different.
1-106 A new "Smith" absolute temperature scale is proposed, and a value of 1000 S is assigned to the boiling point of water. The ice point on this scale, and its relation to the Kelvin scale are to be determined. Analysis All linear absolute temperature scales read zero at absolute zero pressure, and are constant multiples of each other. For example, T(R) = 1.8 T(K). That is, multiplying a temperature value in K by 1.8 will give the same temperature in R. The proposed temperature scale is an acceptable absolute temperature scale since it differs from the other absolute temperature scales by a constant only. The boiling temperature of water in the Kelvin and the Smith scales are 315.15 K and 1000 K, respectively. Therefore, these two temperature scales are related to each other by
T (S) 1000 T ( K) 373.15 2.6799 T(K )
0 K 373.15 S 1000
The ice point of water on the Smith scale is T(S)ice = 2.6799 T(K)ice = 2.6799 273.15 = 732.0 S
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1-107E An expression for the equivalent wind chill temperature is given in English units. It is to be converted to SI units. Analysis The required conversion relations are 1 mph = 1.609 km/h and T( F) = 1.8T( C) + 32. The first thought that comes to mind is to replace T( F) in the equation by its equivalent 1.8T( C) + 32, and V in mph by 1.609 km/h, which is the "regular" way of converting units. However, the equation we have is not a regular dimensionally homogeneous equation, and thus the regular rules do not apply. The V in the equation is a constant whose value is equal to the numerical value of the velocity in mph. Therefore, if V is given in km/h, we should divide it by 1.609 to convert it to the desired unit of mph. That is,
Tequiv ( F) 914 [914 Tambient ( F)][0.475 0.0203(V / 1609) 0.304 V / 1609 ] . . . .
or
Tequiv ( F) 914 [914 Tambient ( F)][0.475 0.0126V . . 0.240 V ]
where V is in km/h. Now the problem reduces to converting a temperature in F to a temperature in C, using the proper convection relation:
18Tequiv ( C) 32 914 [914 (18Tambient ( C) 32)][0.475 0.0126V . . . . 0.240 V ]
which simplifies to
Tequiv ( C) 33.0 (33.0 Tambient )(0.475 0.0126V 0.240 V )
where the ambient air temperature is in C.
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1-108E EES Problem 1-107E is reconsidered. The equivalent wind-chill temperatures in F as a function of wind velocity in the range of 4 mph to 100 mph for the ambient temperatures of 20, 40, and 60F are to be plotted, and the results are to be discussed. Analysis The problem is solved using EES, and the solution is given below.
"Obtain V and T_ambient from the Diagram Window" {T_ambient=10 V=20} V_use=max(V,4) T_equiv=91.4-(91.4-T_ambient)*(0.475 - 0.0203*V_use + 0.304*sqrt(V_use)) "The parametric table was used to generate the plot, Fill in values for T_ambient and V (use Alter Values under Tables menu) then use F3 to solve table. Plot the first 10 rows and then overlay the second ten, and so on. Place the text on the plot using Add Text under the Plot menu." Tequiv [F] -52 -46 -40 -34 -27 -21 -15 -9 -3 3 -75 -68 -61 -53 -46 -39 -32 -25 -18 -11 -87 -79 -72 -64 -56 -49 -41 -33 -26 -18 -93 -85 -77 -69 -61 -54 -46 -38 -30 -22 Tambient [F] -25 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 -25 -20 -15 -10 -5 0 5 10 15 20 V [mph] 10 10 10 10 10 10 10 10 10 10 20 20 20 20 20 20 20 20 20 20 30 30 30 30 30 30 30 30 30 30 40 40 40 40 40 40 40 40 40 40
60 50 40 30
Tamb = 60F
T equiv [F]
20
Tamb = 40F
10 0 -10
Tamb = 20F
-20 0 20 40 60 80 100
V [mph]
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1-109 One section of the duct of an air-conditioning system is laid underwater. The upward force the water will exert on the duct is to be determined. Assumptions 1 The diameter given is the outer diameter of the duct (or, the thickness of the duct material is negligible). 2 The weight of the duct and the air in is negligible. Properties The density of air is given to be = 1.30 kg/m3. We take the density of water to be 1000 kg/m 3. Analysis Noting that the weight of the duct and the air in it is negligible, the net upward force acting on the duct is the buoyancy force exerted by water. The volume of the underground section of the duct is
V AL ( D 2 / 4) L [ (0.15 m) 2 /4](20 m) = 0.353 m 3
D =15 cm L = 20 m FB
Then the buoyancy force becomes
FB gV (1000 kg/m 3 )(9.81 m/s 2 )(0.353 m 3 ) 1 kN 1000 kg m/s 2 3.46 kN
Discussion The upward force exerted by water on the duct is 3.46 kN, which is equivalent to the weight of a mass of 353 kg. Therefore, this force must be treated seriously.
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1-110 A helium balloon tied to the ground carries 2 people. The acceleration of the balloon when it is first released is to be determined. Assumptions The weight of the cage and the ropes of the balloon is negligible. Properties The density of air is given to be = 1.16 kg/m3. The density of helium gas is 1/7th of this.
Analysis The buoyancy force acting on the balloon is
V balloon
FB
4 r 3 /3 4(5 m) 3 /3 523.6 m 3 (1.16 kg/m )(9.81m/s 2 )(523.6 m 3 )
air gV balloon 3
1N 1 kg m/s 2
5958 N
The total mass is
m He m total
HeV
m He
1.16 kg/m 3 (523.6 m 3 ) 86.8 kg 7 m people 86.8 2 70 226.8 kg
The total weight is
W m total g (226.8 kg)(9.81 m/s 2 ) 1N 1 kg m/s 2 2225 N
Thus the net force acting on the balloon is
Fnet FB W 5958 2225 3733 N
Then the acceleration becomes
a Fnet m total 3733 N 226.8 kg 1kg m/s 2 1N 16.5 m/s 2
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1-111 EES Problem 1-110 is reconsidered. The effect of the number of people carried in the balloon on acceleration is to be investigated. Acceleration is to be plotted against the number of people, and the results are to be discussed. Analysis The problem is solved using EES, and the solution is given below. "Given Data:" rho_air=1.16"[kg/m^3]" "density of air" g=9.807"[m/s^2]" d_balloon=10"[m]" m_1person=70"[kg]" {NoPeople = 2} "Data suppied in Parametric Table" "Calculated values:" rho_He=rho_air/7"[kg/m^3]" "density of helium" r_balloon=d_balloon/2"[m]" V_balloon=4*pi*r_balloon^3/3"[m^3]" m_people=NoPeople*m_1person"[kg]" m_He=rho_He*V_balloon"[kg]" m_total=m_He+m_people"[kg]" "The total weight of balloon and people is:" W_total=m_total*g"[N]" "The buoyancy force acting on the balloon, F_b, is equal to the weight of the air displaced by the balloon." F_b=rho_air*V_balloon*g"[N]" "From the free body diagram of the balloon, the balancing vertical forces must equal the product of the total mass and the vertical acceleration:" F_b- W_total=m_total*a_up
2
Aup [m/s ] 28.19 16.46 10.26 6.434 3.831 1.947 0.5204 -0.5973 -1.497 -2.236
NoPeople 1 2 3 4 5 6 7 8 9 10
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1-112 A balloon is filled with helium gas. The maximum amount of load the balloon can carry is to be determined. Assumptions The weight of the cage and the ropes of the balloon is negligible. Properties The density of air is given to be The density of helium gas is 1/7th of this. = 1.16 kg/m3.
Analysis In the limiting case, the net force acting on the balloon will be zero. That is, the buoyancy force and the weight will balance each other:
W m total mg FB g FB 5958 N 9.81 m/s 2 607.3 kg
Helium balloon
Thus,
mpeople mtotal mHe 607.3 86.8 520.5 kg
m
1-113E The pressure in a steam boiler is given in kgf/cm 2. It is to be expressed in psi, kPa, atm, and bars. Analysis We note that 1 atm = 1.03323 kgf/cm2, 1 atm = 14.696 psi, 1 atm = 101.325 kPa, and 1 atm = 1.01325 bar (inner cover page of text). Then the desired conversions become: In atm:
P (92 kgf/cm2 ) 1 atm 1.03323 kgf/cm2 1 atm 1.03323 kgf/cm
2
89.04 atm
In psi:
P
(92 kgf/cm 2 )
14.696 psi 1 atm
101.325 kPa 1 atm
1309 psi
In kPa:
P
(92 kgf/cm 2 )
1 atm 1.03323 kgf/cm
2
9022 kPa
In bars:
P
(92 kgf/cm 2 )
1 atm 1.03323 kgf/cm
2
1.01325 bar 1 atm
90.22 bar
Discussion Note that the units atm, kgf/cm2, and bar are almost identical to each other.
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1-114 A 10-m high cylindrical container is filled with equal volumes of water and oil. The pressure difference between the top and the bottom of the container is to be determined. Properties The density of water is given to be The specific gravity of oil is given to be 0.85. = 1000 kg/m3.
Oil SG = 0.85 h = 10 m Water
Analysis The density of the oil is obtained by multiplying its specific gravity by the density of water,
SG
H 2O
(0.85)(1000 kg/m 3 ) 850 kg/m 3
The pressure difference between the top and the bottom of the cylinder is the sum of the pressure differences across the two fluids,
Ptotal Poil Pwater
3
( gh) oil
2
( gh) water
(850 kg/m )(9.81 m/s )(5 m) (1000 kg/m 3 )(9.81 m/s 2 )(5 m) 90.7 kPa
1 kPa 1000 N/m 2
1-115 The pressure of a gas contained in a vertical piston-cylinder device is measured to be 250 kPa. The mass of the piston is to be determined. Assumptions There is no friction between the piston and the cylinder. Analysis Drawing the free body diagram of the piston and balancing the vertical forces yield
W mg (m)(9.81 m/s 2 ) PA Patm A ( P Patm ) A
Patm
P W = mg
1000 kg/m s 2 (250 100 kPa)(30 10 4 m 2 ) 1kPa
It yields
m = 45.9 kg
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1-116 The gage pressure in a pressure cooker is maintained constant at 100 kPa by a petcock. The mass of the petcock is to be determined. Assumptions There is no blockage of the pressure release valve. Analysis Atmospheric pressure is acting on all surfaces of the petcock, which balances itself out. Therefore, it can be disregarded in calculations if we use the gage pressure as the cooker pressure. A force balance on the petcock ( Fy = 0) yields
W m Pgage A Pgage A g 0.0408 kg (100 kPa)(4 10 9.81 m/s 2
6
Patm
P W = mg
m 2 ) 1000 kg/m s 2 1 kPa
1-117 A glass tube open to the atmosphere is attached to a water pipe, and the pressure at the bottom of the tube is measured. It is to be determined how high the water will rise in the tube. Properties The density of water is given to be = 1000 kg/m3.
Analysis The pressure at the bottom of the tube can be expressed as
P Patm ( g h) tube
h
Solving for h,
h P Patm g (115 92) kPa (1000 kg/m )(9.81 m/s ) 2.34 m
3 2
Patm= 92 kPa Water
1 kg m/s 1N
2
1000 N/m 1 kPa
2
1-118 The average atmospheric pressure is given as Patm 101.325(1 0.02256 z )5.256 where z is the altitude in km. The atmospheric pressures at various locations are to be determined. Analysis The atmospheric pressures at various locations are obtained by substituting the altitude z values in km into the relation
5.256 P . atm 101325(1 0.02256z)
Atlanta: Denver: M. City: Mt. Ev.:
(z = 0.306 km): Patm = 101.325(1 - 0.02256 0.306)5.256 = 97.7 kPa (z = 1.610 km): Patm = 101.325(1 - 0.02256 1.610)5.256 = 83.4 kPa (z = 2.309 km): Patm = 101.325(1 - 0.02256 2.309)5.256 = 76.5 kPa (z = 8.848 km): Patm = 101.325(1 - 0.02256 8.848)5.256 = 31.4 kPa
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1-119 The air pressure in a duct is measured by an inclined manometer. For a given vertical level difference, the gage pressure in the duct and the length of the differential fluid column are to be determined. Assumptions The manometer fluid is an incompressible substance. Properties The density of the liquid is given to be = 0.81 kg/L = 810 kg/m3. Analysis The gage pressure in the duct is determined from
Pgage Pabs Patm gh 1N 1 kg m/s
2
(810 kg/m 3 )(9.81 m/s 2 )(0.08 m) 636 Pa
1 Pa 1 N/m 2
The length of the differential fluid column is
L h / sin
(8 cm) / sin 35
13.9 cm
Discussion Note that the length of the differential fluid column is extended considerably by inclining the manometer arm for better readability.
1-120E Equal volumes of water and oil are poured into a U-tube from different arms, and the oil side is pressurized until the contact surface of the two fluids moves to the bottom and the liquid levels in both arms become the same. The excess pressure applied on the oil side is to be determined. Assumptions 1 Both water and oil are incompressible substances. 2 Oil does not mix with water. 3 The cross-sectional area of the U-tube is constant. Properties The density of oil is given to be oil = 49.3 lbm/ft3. We take the density of water to be w = 62.4 lbm/ft3. Analysis Noting that the pressure of both the water and the oil is the same at the contact surface, the pressure at this surface can be expressed as
Pcontact Pblow
a gha
Patm
w gh w
Noting that ha = hw and rearranging,
Pgage,blow Pblow Patm (
w oil ) gh
(62.4 - 49.3 lbm/ft 3 )(32.2 ft/s 2 )(30/12 ft) 0.227 psi
1 lbf 32.2 lbm ft/s 2
1 ft 2 144 in 2
Discussion When the person stops blowing, the oil will rise and some water will flow into the right arm. It can be shown that when the curvature effects of the tube are disregarded, the differential height of water will be 23.7 in to balance 30-in of oil.
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1-121 It is given that an IV fluid and the blood pressures balance each other when the bottle is at a certain height, and a certain gage pressure at the arm level is needed for sufficient flow rate. The gage pressure of the blood and elevation of the bottle required to maintain flow at the desired rate are to be determined. Assumptions 1 The IV fluid is incompressible. 2 The IV bottle is open to the atmosphere. Properties The density of the IV fluid is given to be = 1020 kg/m3. Analysis (a) Noting that the IV fluid and the blood pressures balance each other when the bottle is 1.2 m above the arm level, the gage pressure of the blood in the arm is simply equal to the gage pressure of the IV fluid at a depth of 1.2 m,
Pgage, arm Pabs Patm gharm - bottle 1 kN 1000 kg m/s
2
(1020 kg/m 3 )(9.81 m/s 2 )(1.20 m) 12.0 k Pa
1 kPa 1 kN/m 2
(b) To provide a gage pressure of 20 kPa at the arm level, the height of the bottle from the arm level is again determined from Pgage, arm gharm -bottle to be
harm - bottle Pgage, arm g 20 kPa (1020 kg/m 3 )(9.81 m/s 2 ) 1000 kg m/s 2 1 kN 1 kN/m 2 1 kPa 2.0 m
Discussion Note that the height of the reservoir can be used to control flow rates in gravity driven flows. When there is flow, the pressure drop in the tube due to friction should also be considered. This will result in raising the bottle a little higher to overcome pressure drop.
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1-122E A water pipe is connected to a double-U manometer whose free arm is open to the atmosphere. The absolute pressure at the center of the pipe is to be determined. Assumptions 1 All the liquids are incompressible. 2 The solubility of the liquids in each other is negligible. Properties The specific gravities of mercury and oil are given to be 13.6 and 0.80, respectively. We take the density of water to be w = 62.4 lbm/ft3. Analysis Starting with the pressure at the center of the water pipe, and moving along the tube by adding (as we go down) or subtracting (as we go up) the gh terms until we reach the free surface of oil where the oil tube is exposed to the atmosphere, and setting the result equal to Patm gives
Pwater pipe
water ghwater oil ghoil Hg ghHg oil ghoil
Patm
Solving for Pwater
pipe,
Pwater pipe
Patm
water g (hwater
SGoil hoil
SGHg hHg
SGoil hoil )
Substituting,
Pwater pipe 14.2 psia (62.4lbm/ft 3 )(32.2 ft/s 2 )[(35/12 ft) 0.8(60/12 ft) 13.6(15/12 ft) 0.8(40/12 ft)] 22.3 psia 1 lbf 32.2 lbm ft/s 2 1 ft 2 144 in 2
Therefore, the absolute pressure in the water pipe is 22.3 psia. Discussion Note that jumping horizontally from one tube to the next and realizing that pressure remains the same in the same fluid simplifies the analysis greatly.
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1-123 The temperature of the atmosphere varies with altitude z as T
2
T0
z , while the gravitational
acceleration varies by g ( z ) g 0 /(1 z / 6,370,320) . Relations for the variation of pressure in atmosphere are to be obtained (a) by ignoring and (b) by considering the variation of g with altitude. Assumptions The air in the troposphere behaves as an ideal gas. Analysis (a) Pressure change across a differential fluid layer of thickness dz in the vertical z direction is dP gdz From the ideal gas relation, the air density can be expressed as
dP P R(T0
z 0
P RT
P R(T0 z)
. Then,
z)
gdz
Separating variables and integrating from z = 0 where P
P P0
P0 to z = z where P = P,
dP P
gdz R(T0 z)
Performing the integrations. T z g P ln ln 0 P0 R T0 Rearranging, the desired relation for atmospheric pressure for the case of constant g becomes
P P0 1 z T0
g R
(b) When the variation of g with altitude is considered, the procedure remains the same but the expressions become more complicated, g0 P dP dz R(T0 z ) (1 z / 6,370,320) 2 Separating variables and integrating from z = 0 where P
P P0
P0 to z = z where P = P,
dP P
z 0
g 0 dz R(T0 z )(1 z / 6,370,320) 2
z
Performing the integrations,
ln P
P P0
g0 1 R (1 kT0 / )(1 kz)
1 kz ln 2 T0 z (1 kT0 / )
1
0
where R = 287 J/kg K = 287 m2/s2 K is the gas constant of air. After some manipulations, we obtain
P P0 exp R( g0 1 kT0 ) 1 1 / kz 1 1 kT0 / ln 1 1 kz z / T0
where T0 = 288.15 K,
= 0.0065 K/m, g0 = 9.807 m/s2, k = 1/6,370,320 m-1, and z is the elevation in m..
Discussion When performing the integration in part (b), the following expression from integral tables is used, together with a transformation of variable x T0 z,
dx x(a bx )
2
1 a(a bx )
1 a
2
ln
a bx x
Also, for z = 11,000 m, for example, the relations in (a) and (b) give 22.62 and 22.69 kPa, respectively.
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1-124 The variation of pressure with density in a thick gas layer is given. A relation is to be obtained for pressure as a function of elevation z. Assumptions The property relation P
C
n
is valid over the entire region considered.
Analysis The pressure change across a differential fluid layer of thickness dz in the vertical z direction is given as,
dP gdz
C
n
Also, the relation P
0 (P /
can be expressed as C
P/
n
P0 /
n 0
, and thus
P0 ) 1 / n
Substituting,
dP g
0 (P /
P0 )1/ n dz
Separating variables and integrating from z = 0 where P
P P0
P0
C
n 0
to z = z where P = P,
( P / P0 )
1/ n
z
dP
0g
dz
0
Performing the integrations.
P0 ( P / P0 ) 1 / n 1/ n 1
1 P 0 gz P0
P P0
( n 1) / n
1
n 1 0 gz n P0
Solving for P,
P P0 1 n 1 0 gz n P0
n /( n 1)
which is the desired relation. Discussion The final result could be expressed in various forms. The form given is very convenient for calculations as it facilitates unit cancellations and reduces the chance of error.
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1-125 A pressure transducers is used to measure pressure by generating analogue signals, and it is to be calibrated by measuring both the pressure and the electric current simultaneously for various settings, and the results are tabulated. A calibration curve in the form of P = aI + b is to be obtained, and the pressure corresponding to a signal of 10 mA is to be calculated. Assumptions Mercury is an incompressible liquid. Properties The specific gravity of mercury is given to be 13.56, and thus its density is 13,560 kg/m 3. Analysis For a given differential height, the pressure can be calculated from
P
g h
For h = 28.0 mm = 0.0280 m, for example,
P 13.56(1000 kg/m 3 )(9.81m/s 2 )(0.0280 m)
Repeating the calculations and tabulating, we have 28.0 3.73 4.21 181.5 24.14 5.78 297.8 39.61 6.97 413.1 54.95 8.15 765.9 101.9 11.76
1 kN 1000 kg m/s 2
1 kPa 1 kN/m 2
3.75 kPa
h(mm ) P(kPa) I (mA)
1027 136.6 14.43
1149 152.8 15.68
1362 181.2 17.86
1458 193.9 18.84
1536 204.3 19.64
A plot of P versus I is given below. It is clear that the pressure varies linearly with the current, and using EES, the best curve fit is obtained to be P = 13.00I - 51.00 (kPa) for 4.21 I
19.64 .
For I = 10 mA, for example, we would get P = 79.0 kPa
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Discussion Note that the calibration relation is valid in the specified range of currents or pressures.
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Fundamentals of Engineering (FE) Exam Problems
1-126 Consider a fish swimming 5 m below the free surface of water. The increase in the pressure exerted on the fish when it dives to a depth of 45 m below the free surface is (a) 392 Pa (b) 9800 Pa (c) 50,000 Pa (d) 392,000 Pa (e) 441,000 Pa
Answer (d) 392,000 Pa
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). rho=1000 "kg/m3" g=9.81 "m/s2" z1=5 "m" z2=45 "m" DELTAP=rho*g*(z2-z1) "Pa" "Some Wrong Solutions with Common Mistakes:" W1_P=rho*g*(z2-z1)/1000 "dividing by 1000" W2_P=rho*g*(z1+z2) "adding depts instead of subtracting" W3_P=rho*(z1+z2) "not using g" W4_P=rho*g*(0+z2) "ignoring z1"
1-127 The atmospheric pressures at the top and the bottom of a building are read by a barometer to be 96.0 and 98.0 kPa. If the density of air is 1.0 kg/m3, the height of the building is (a) 17 m Answer (d) 204 m (b) 20 m (c) 170 m (d) 204 m (e) 252 m
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). rho=1.0 "kg/m3" g=9.81 "m/s2" P1=96 "kPa" P2=98 "kPa" DELTAP=P2-P1 "kPa" DELTAP=rho*g*h/1000 "kPa" "Some Wrong Solutions with Common Mistakes:" DELTAP=rho*W1_h/1000 "not using g" DELTAP=g*W2_h/1000 "not using rho" P2=rho*g*W3_h/1000 "ignoring P1" P1=rho*g*W4_h/1000 "ignoring P2"
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1-128 An apple loses 4.5 kJ of heat as it cools per C drop in its temperature. The amount of heat loss from the apple per F drop in its temperature is (a) 1.25 kJ (b) 2.50 kJ (c) 5.0 kJ (d) 8.1 kJ (e) 4.1 kJ Answer (b) 2.50 kJ
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). Q_perC=4.5 "kJ" Q_perF=Q_perC/1.8 "kJ" "Some Wrong Solutions with Common Mistakes:" W1_Q=Q_perC*1.8 "multiplying instead of dividing" W2_Q=Q_perC "setting them equal to each other"
1-129 Consider a 2-m deep swimming pool. The pressure difference between the top and bottom of the pool is (a) 12.0 kPa Answer (b) 19.6 kPa (b) 19.6 kPa (c) 38.1 kPa (d) 50.8 kPa (e) 200 kPa
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). rho=1000 "kg/m^3" g=9.81 "m/s2" z1=0 "m" z2=2 "m" DELTAP=rho*g*(z2-z1)/1000 "kPa" "Some Wrong Solutions with Common Mistakes:" W1_P=rho*(z1+z2)/1000 "not using g" W2_P=rho*g*(z2-z1)/2000 "taking half of z" W3_P=rho*g*(z2-z1) "not dividing by 1000"
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1-130 At sea level, the weight of 1 kg mass in SI units is 9.81 N. The weight of 1 lbm mass in English units is (a) 1 lbf Answer (a) 1 lbf (b) 9.81 lbf (c) 32.2 lbf (d) 0.1 lbf (e) 0.031 lbf
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). m=1 "lbm" g=32.2 "ft/s2" W=m*g/32.2 "lbf" "Some Wrong Solutions with Common Mistakes:" gSI=9.81 "m/s2" W1_W= m*gSI "Using wrong conversion" W2_W= m*g "Using wrong conversion" W3_W= m/gSI "Using wrong conversion" W4_W= m/g "Using wrong conversion"
1-131 During a heating process, the temperature of an object rises by 20 C. This temperature rise is equivalent to a temperature rise of (a) 20 F Answer (d) 36 R (b) 52 F (c) 36 K (d) 36 R (e) 293 K
Solution Solved by EES Software. Solutions can be verified by copying-and-pasting the following lines on a blank EES screen. (Similar problems and their solutions can be obtained easily by modifying numerical values). T_inC=20 "C" T_inR=T_inC*1.8 "R" "Some Wrong Solutions with Common Mistakes:" W1_TinF=T_inC "F, setting C and F equal to each other" W2_TinF=T_inC*1.8+32 "F, converting to F " W3_TinK=1.8*T_inC "K, wrong conversion from C to K" W4_TinK=T_inC+273 "K, converting to K"
1-132 ... 1-134 Design, Essay, and Experiment Problems
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PROPRIETARY MATERIAL. 2008 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course preparation. If you are a student using this Manual, you are using it without permission.