Symbolic Solution
Problem 6.21 A cantilever beam
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Chapter 6 Fundamental Dimensions and Units Those of you who will pursue aerospace, chemical, mechanical, or materials engineering will learn about the underlying concepts that lead to the solution in your heat-transfer class. What is the appropriate unit forhif the preceding equation is to be homogeneous in units? Show all steps of your work.6.24. The air resistance to the motion of a vehicle is some- thing important that engineers investigate. As you may also know, the drag force acting on a car is determined experimentally by placing the car in a wind tunnel. For a given car, the experimental data is generally repre- sented by a single coefficient that is called the drag coefficient. It is given by the following relationship:
where
Cddrag coefficient
Fdmeasured drag force (lb) rair density (slugs/ft3)
Vair speed inside the wind tunnel (ft/s) A frontal area of the car (ft2)
What is the appropriate unit forCdif the preceding equation is to be homogeneous in units? Show all steps of your work.
6.25. Fins, or extended surfaces, commonly are used in a variety of engineering applications to enhance cooling.
Common examples include a motorcycle engine head, a lawn mower engine head, heat sinks used in elec- tronic equipment, and finned tube heat exchangers in room heating and cooling applications. For long fins, the temperature distribution along the fin is given by the exponential relationship:
where
Ttemperature (K)
h the heat transfer coefficient (W/m2ⴢK) pperimeter of the fin (m)
mB
hp kA
TTambient 1TbaseTambient2emx Cd Fd
1 2rV 2A
A cross-sectional area of the fin (m2) kthermal conductivity of the fin material x distance from the base of the fin (m) What is the appropriate unit fork if the preceding equation is to be homogeneous in units? Show all steps of your work.
6.26. A person who is 180 cm tall and weights 750 newtons is driving a car at a speed of 90 kilometers per hour over a distance of 80 kilometers. The outside air tem- perature is 30°C and has a density of 1.2 (kg/m3) kilo- grams per cubic meter. Convert all of the values given in this example from SI units to U.S. Customary units.
6.27. Use the conversion factors given on the front and back end covers of this text and convert the given values.
(a) Value of area A16 in2to ft2, (b) Value of volume V64 in3to ft3, (c) Value of area moment of inertia I21.3 in4to ft4.
6.28. The acceleration due to gravity gis 9.81 m/s2. Express the value gin U.S. Customary and B.G. units. Show all conversion steps.
6.29. Sir Isaac Newton discovered that any 2 masses m1and m2attract each other with a force that is equal in mag- nitude and acts in the opposite direction, according to the following relationship:
where
Fattractive force, N
GUniversal Gravitational Constant m1mass of particle-1, kg
m2mass of particle-2, kg
rdistance between the center of each particle, m
What is the appropriate unit for G, if the above equa- tion is to be homogeneous in units?
6.30. Convert the atmospheric pressure in the given units to requested units. Show all the conversion steps. (a) 14.7 lbf/in2to lbf/ft2, (b) 14.7 lbf/in2to Pa, (c) 14.7 lbf/in2to kPa, (d) 14.7 lbf/in2to bar (1 bar 100 kPa).
6.31. For gases under certain conditions, there is a relation- ship between the pressure of the gas, its volume, and its
F Gm1m2
r2
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temperature as given by what is commonly called the ideal gas law. The ideal gas law is given by
PVmRT
where
Pabsolute pressure of the gas, Pa Vvolume of the gas, m3
mmass, kg Rgas constant
Tabsolute temperature, Kelvin
What is the appropriate unit for R, if the above equa- tion is to be homogeneous in units?
6.32. The amount of radiant energy emitted by a surface is given by the equation whereqrepresents the rate of thermal energy, per unit time, emitted by the surface in watts;eis the emissivity of the surface 0e 1 and is unitless;sis the Stefan-Boltzman constant (s5.67108);Arepresents the area of the surface in m2; andTSis the surface temperature of the object expressed in Kelvin. What is the appropriate unit fors, if the above equation is to be homogeneous in units?
6.33. A person’s body temperature is controlled by (1) convec- tive and radiative heat transfer to the surroundings, (2) sweating, (3) respiration by breathing surrounding air and exhaling it at approximately body temperature, (4) blood circulation near the surface of the skin, and (5) metabolic rate. Metabolic rate determines the rate of conversion of chemical to thermal energy within a per- son’s body. The metabolic rate depends on the person’s activity level. A unit commonly used to express the meta- bolic rate for an average person under sedentary condi- tions, per unit surface area, is called met; 1 met is equal to 58.2 W/m2. Convert this value to Btu/hft2. Also, calculate the amount of energy dissipated by an average adult person sleeping for 8 hours, if he or she generates 0.7 mets and has a body surface area of 19.6 ft2. Express your results in Btu and Joules (1 Btu 1055 Joules).
6.34. Thecalorieis defined as the amount of heat required to raise the temperature of 1 gram of water by 1°C. Also, the energy content of food is typically expressed inCalories which is equal to 1000 calories. Convert the results of the previous problem to Calories (1 Btu252 calories).
6.35. Convert the strength of selected materials given in the accompanying table from MPa to ksi, where 1000 lbf/in21 ksi.
qesAT4S,
6.36. The density of water is 1000 kg/m3. Express the density of water in lbm/ft3and lbm/gallon (7.48 gallons 1 ft3).
6.37. A unit that is generally used to express the insulating value of clothing is called clo. 1 clo is equal to 0.155 m2 °C / W. Express this value in U.S. Custom- ary units (°Fft2h/Btu).
6.38. Start with 1 lbfft/s and convert it to Nm/s and show that 1 lbfft /s is equal to 1.36 watts. Knowing that 550 lbfft /s is equal to 1 horsepower, how many kilowatts is that?
6.39. Viscosity of fluid plays a significant role in the analyses of many fluid dynamics problems. The viscosity of water can be determined from the following correlation:
where
mviscosity (N/sm2) Ttemperature (K) c12.414 105 c2247.8 (K) c3140 (K)
What is the appropriate unit for c1, if the above equa- tion is to be homogeneous in units?
6.40. For the ideal gas law given in Problem 6.31, if the units of P, V, m, and Tare expressed in lbf/ft2, ft3, lbm, and
a c2
Tc3b
mc110
#
Problems
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Ultimate Ultimate strength strength
Material (MPa) (ksi)
Aluminum alloys 100 –550
Concrete (compression) 10 –70 Steel
Machine 550 –860
Spring 700 –1,900
Stainless 400 –1,000
Tool 900
Structural Steel 340 –830 Titanium alloys 900 –1,200 Wood (Bending)
Douglas fir 50 –80
Oak 50 –100
Southern pine 50 –100
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Chapter 6 Fundamental Dimensions and Units degree Rankine (°R), respectively, what is the appro- priate unit for gas constant Rif the ideal gas law is to be homogeneous in units?6.41. For the fin equation described in Problem 6.25, if the units of T, h, P, A, and x are expressed in degree Rankine (°R), Btu /hft2°R, ft, ft2, and ft, respec- tively, what is the appropriate unit for thermal con- ductivity kif the fin equation is to be homogeneous in unit? Show all steps of your work.
6.42. For Example 6.3 part (a), if the units of d, P, L, and A are given in inches, lbf, inches, and in2, respectively, what are the units for modulus of elasticity E?
6.43. For Example 6.3 part (b), if the units of k, A, T1T2, and Lare given in Btu/hft°F, ft2,°F, and ft, respec- tively, what is the appropriate unit for heat transfer rate q?
6.44. For the drag coefficient relationship given in
Problem 6.24, if the units of Fd, V, A, and Cd, are expressed in N, m/s, m2, and unitless, respectively, what is the appropriate unit for ?
6.45. For the cantilever beam relationship given in Problem 6.21, if the units of y, w, E, x, and Lare expressed in inches, lbf/in, lbf/in2, inches, and inches, respectively, what is the appropriate unit for I?
6.46. The rotation of a rigid object is governed by the
following relationship: MI, where Msum of the moments due to external forces (Nm), I mass moment of inertia, and angular acceleration of the object (rad /s2). What is the appropriate unit for mass moment of inertia I?
6.47. The value of viscosity of a fluid represents a measure of how easily the given fluid can flow. The higher the vis- cosity value is, the more resistance the fluid offers to flow. For example, it would require less energy to trans- port water in a pipe than it would to transport motor oil or glycerin. The viscosity of many fluids is governed by Newton’s law of viscosity where is shear stress (N/m2), is viscosity, and duis change in flow speed (m/s) over a height dy(m). What is the appropriate unit for viscosity?
6.48. The power output of a water turbine is given by P gQh, where Pis power, is density of water (kg/m3), gis acceleration due to gravity (m/s2), Qis water flow rate (m3/s), and his the available water head (m). What is the appropriate unit for P?
6.49. The head loss due to flow of a fluid inside a pipe is calculated from , where hLoss(m), fis friction factor, Lis pipe length (m), Dis pipe diameter (m), Vis average flow velocity (m/s), and gis the accel- eration due to gravity (m/s2). What is the appropriate unit for friction factor f?
6.50. For Problem 6.49, hLossis expressed in feet, Land Din inches, Vin (ft/s), and gin (ft/s2). What is the appro- priate unit for the friction factor f?
hLossf L D V2
2g tm du
dy ,
“Nothing is too wonderful to be true if it be consistent with the laws of nature.”
—Michael Faraday (1791–1867)
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