Problem 1.6 Make a guess at the order of magnitude of the mass (e.g., 0.01, 0.1, 1.0, 10, 100, or 1000 lbm or kg) of standard air that is in a room 10

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6 Problem 1.6 Make a guess at the order of magnitude of the mass (e.g., 0.01, 0.1, 1.0, 10, 100, or 1000 lbm or kg) of standard air that is in a room 10 ft by 10 ft by 8 ft, and then compute this mass in lbm and kg to see how close your estimate was. Solution Given: Dimensions of a room. Find: Mass of air in lbm and kg. The data for standard air are: R air ft lbf lbmr p 14.7psi T ( ) R 519R Then ρ p R air T ρ 14.7 lbf in lbmr ftlbf 1 519R 12in 1ft 2 ρ lbm ft 3 or ρ 1.23 kg m 3 The volume of the room is V 10ft 10ft 8ft V 800ft 3 The mass of air is then m ρv m lbm ft 3 800ft 3 m 61.2 lbm m 27.8 kg

7 Problem 1.7 Given: Data on nitrogen tank Find: Mass of nitrogen Solution The given or available data is: D 6in L 4.25ft p 204atm T ( ) R T 519R R N ft lbf lbr (Table A.6) The governing equation is the ideal gas equation p ρr N2 T and ρ M V where V is the tank volume V π 4 D2 L V π V 0.834ft 3 12 ft 4.25ft Hence M Vρ pv R N2 T lbf M in 2 144in 2 ft ft lbr ftlbf R lbft s 2 lbf M 12.6 lb M slug

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16 Problem 1.16 From Appendix A, the viscosity µ (N. s/m 2 ) of water at temperature T (K) can be computed from µ A10 B/(T - C), where A X 10-5 N.s/m 2, B K, and C 140 K. Determine the viscosity of water at 20 C, and estimate its uncertainty if the uncertainty in temperature measurement is +/ C. Solution Given: Data on water. Find: Viscosity and uncertainty in viscosity. The data provided are: A Ns m 2 B 247.8K C 140K T 293K The uncertainty in temperature is u T 0.25K 293K u T 0.085% The formula for viscosity is B ( T C) µ ( T) A10 Evaluating µ 247.8K µ ( T) Ns ( 293K 140K) m 2 10 µ ( T) Ns m 2 For the uncertainty B d dt µ ( T) ( T C) B A10 ( T C) 2 ln( 10)

17 so u µ ( T) T d µ ( T) T µ ( T) u B T ln( 10) T d ( T C) 2 u T Using the given data 247.8K u µ ( T) ln( 10) 293K ( 293K 140K) % u µ ( T) 0.61%

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21 Problem 1.20 The height of a building may be estimated by measuring the horizontal distance to a point on t ground and the angle from this point to the top of the building. Assuming these measurements L 100 +/- 0.5 ft and θ 30 +/- 0.2 degrees, estimate the height H of the building and the uncertainty in the estimate. For the same building height and measurement uncertainties, use Excel s Solver to determine the angle (and the corresponding distance from the building) at which measurements should be made to minimize the uncertainty in estimated height. Evaluat and plot the optimum measurement angle as a function of building height for 50 < H < 1000 f Solution Given: Data on length and angle measurements. Find: The data provided are: L 100ft δl 0.5ft θ 30deg δθ 0.2deg The uncertainty in L is u L δl L u L 0.5% The uncertainty in θ is u θ δθ θ u θ 0.667% The height H is given by H Ltan( θ) H 57.7 ft For the uncertainty u H L H L Hu L 2 + θ H θ Hu θ 2

22 and L H tan( θ) θ H ( ( ) 2 ) L 1 + tan θ ( ) L so u H H tan θ u L 2 + ( ) Lθ H 1 + tan( θ)2 2 u θ Using the given data 100 u H 57.5 tan π π tan π u H 0.95% δh u H H δh 0.55 ft H ft The angle θ at which the uncertainty in H is minimized is obtained from the corresponding Exce workbook (which also shows the plot of u H vs θ) θ optimum 31.4deg

23 Problem 1.20 (In Excel) The height of a building may be estimated by measuring the horizontal distance to a point on the ground and the angle from this point to the top of the building. Assuming these measurements are L 100 +/- 0.5 ft and θ 30 +/- 0.2 degrees, estimate the height H of the building and the uncertainty in the estimate. For the same building height and measurement uncertainties, use Excel s Solver to determine the angle (and the corresponding distance from the building) at which measurements should be made to minimize the uncertainty in estimated height. Evaluate and plot the optimum measurement angle as a function of building height for 50 < H < 1000 ft. Given: Data on length and angle measurements. Find: Height of building; uncertainty; angle to minimize uncertainty Given data: H 57.7 ft δl 0.5 ft δθ 0.2 deg For this building height, we are to vary θ (and therefore L ) to minimize the uncertainty u H. ( ( ) 2 ) 2 The uncertainty is L u H H tan ( θ ) Lθ u L + H 1 + tan θ 2 u θ Expressing u H, u L, u θ and L as functions of θ, (remember that δl and δθ are constant, so as L and θ vary the uncertainties will too!) and simplifying Plotting u H vs θ ( ) tan( θ) δl 2 H u H θ ( ( ) 2 ) 1 + tan θ + tan( θ) 2 δθ θ (deg) u H % % % % % % % % % % % % % % % % % Optimizing using Solver u H 12% 10% 8% 6% 4% 2% 0% Uncertainty in Height (H 57.7 ft) vs θ θ ( o )

24 θ (deg) u H % To find the optimum θ as a function of building height H we need a more complex Solver H (ft) θ (deg) u H % % % % % % % % % % % % % % % θ (deg) Optimum Angle vs Building Height H (ft) Use Solver to vary ALL θ's to minimize the total u H! Total u H 's: 11.32%

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29 Problem 1.24 For a small particle of aluminum (spherical, with diameter d mm) falling in standard air at speed V, the drag is given by F D 3πµVd, where µ is the air viscosity. Find the maximum speed starting from rest, and the time it takes to reach 95% of this speed. Plot the speed as a function of time. Solution Given: Data on sphere and formula for drag. Find: Maximum speed, time to reach 95% of this speed, and plot speed as a function of time. The data provided, or available in the Appendices, are: ρ air 1.17 kg m 3 µ Ns m 2 ρ w 999 kg m 3 SG Al 2.64 d 0.025mm Then the density of the sphere is ρ Al SG Al ρ w ρ Al 2637 kg m 3 The sphere mass is πd 3 M ρ Al kg m 3 π ( m) 3 6 M kg Newton's 2nd law for the steady state motion becomes Mg 3π Vd so V max Mg 3 πµ d 1 3 π kg m s s 2 m Ns m

30 V max m s Newton's 2nd law for the general motion is M dv dt Mg 3 πµ Vd so g dv 3 πµ d V m dt Integrating and using limits Vt () Mg 3 πµ d 3 πµ d t M 1 e Using the given data 0.06 V (m/s) t (s) The time to reach 95% of maximum speed is obtained from Mg 3 πµ d 3 πµ d t M 1 e 0.95V max so t M 0.95V max 3 πµ d ln 1 Substituting values t s 3 πµ d Mg

31 Problem 1.24 (In Excel) For a small particle of aluminum (spherical, with diameter d mm) falling in standard air at speed V, the drag is given by F D 3πµVd, where µ is the air viscosity. Find the maximum speed starting from rest, and the time it takes to reach 95% of this speed. Plot the speed as a function of time. Solution Given: Find: Data and formula for drag. Maximum speed, time to reach 95% of final speed, and plot. The data given or availabke from the Appendices is µ 1.80E-05 Ns/m 2 ρ 1.17 kg/m 3 SG Al 2.64 ρ w 999 kg/m 3 d mm Data can be computed from the above using the following equations ρ Al SG Al ρ w Speed V vs Time t π d 3 M ρ Al 6 Mg V max 3 π µ d 3 πµ d Mg t M Vt () 1 e 3 π µ d t (s) V (m/s) ρ Al 2637 kg/m M 2.16E-11 kg Vmax m/s For the time at which V (t ) 0.95V max, use Goal Seek : t (s) V (m/s) 0.95V max Error (%) % V (m/s) t (s)

32 Problem 1.25 For small spherical water droplets, diameter d, falling in standard air at speed V, the drag is given by F D 3πµVd, where µ is the air viscosity. Determine the diameter d of droplets that take 1 second to fall from rest a distance of 1 m. (Use Excel s Goal Seek.) Solution Given: Data on sphere and formula for drag. Find: Diameter of water droplets that take 1 second to fall 1 m. The data provided, or available in the Appendices, are: µ Ns m 2 ρ w 999 kg m 3 Newton's 2nd law for the sphere (mass M) is M dv dt Mg 3 πµ Vd so g dv 3 πµ d V m dt Integrating and using limits Vt () Mg 3 πµ d 3 πµ d t M 1 e Integrating again xt () Mg 3 πµ d t + M 3 πµ d 3 πµ d t M e 1

33 πd 3 Replacing M with an expression involving diameter d M ρ w 6 xt () ρ w d 2 g 18 µ t + ρ w d 2 18 µ 18µ ρ w d 2 t e 1 This equation must be solved for d so that x( 1s ) 1m. The answer can be obtained from manual iteration, or by using Excel's Goal Seek. d 0.193mm 1 x (m) t (s)

34 Problem 1.25 (In Excel) For small spherical water droplets, diameter d, falling in standard air at speed V, the drag is given by F D 3πµVd, where µ is the air viscosity. Determine the diameter d of droplets that take 1 second to fall from rest a distance of 1 m. (Use Excel s Goal Seek.) speed. Plot the speed as a function of time. Solution Given: Data and formula for drag. Find: Diameter of droplets that take 1 s to fall 1 m. The data given or availabke from the Appendices is µ 1.80E-05 Ns/m 2 ρ w 999 kg/m 3 Make a guess at the correct diameter (and use Goal Seek later): (The diameter guess leads to a mass.) d mm M 3.78E-09 kg Data can be computed from the above using the following equations: π d 3 M ρ w 6 xt () Mg t + 3 π µ d M e 3 π µ d 3 πµ d t M 1 Use Goal Seek to vary d to make x (1s) 1 m: t (s) x (m) t (s) x (m) Distance x vs Time t t (s) x (m)

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40 Problem 1.30 Derive the following conversion factors: (a) Convert a pressure of 1 psi to kpa. (b) Convert a volume of 1 liter to gallons. (c) Convert a viscosity of 1 lbf.s/ft 2 to N.s/m 2. Solution Using data from tables (e.g. Table G.2) (a) 1psi 6895Pa 1kPa 1psi 6.89kPa 1psi 1000Pa (b) 1liter 1quart 1gal 1liter 0.264gal 0.946liter 4quart (c) 1 lbf s ft lbf s 4.448N 12 ft ft Ns 1lbf m m 2

41 Problem 1.31 Derive the following conversion factors: (a) Convert a viscosity of 1 m 2 /s to ft 2 /s. (b) Convert a power of 100 W to horsepower. (c) Convert a specific energy of 1 kj/kg to Btu/lbm. Solution Using data from tables (e.g. Table G.2) (a) 1 m2 s m2 12 ft ft2 s m s (b) 100W 1hp 100W 0.134hp 746W (c) 1 kj kg 1 kj 1000J 1Btu 0.454kg 0.43 Btu kg 1kJ 1055J 1lbm lbm

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43 Problem 1.33 Derive the following conversion factors: (a) Convert a volume flow rate in in. 3 /min to mm 3 /s. (b) Convert a volume flow rate in cubic meters per second to gpm (gallons per minute). (c) Convert a volume flow rate in liters per minute to gpm (gallons per minute). (d) Convert a volume flow rate of air in standard cubic feet per minute (SCFM) to cubic meters per hour. A standard cubic foot of gas occupies one cubic foot at standard temperature and pressure (T 15 C and p kpa absolute). Solution Using data from tables (e.g. Table G.2) (a) 1 in3 min 1 in m 1000mm 1min 273 mm3 min 1in 1m 60s s (b) 1 m3 s 1 m3 1quart 1gal 60s s m gpm 4quart 1min (c) 1 liter min 1 liter 1quart 1gal 60s gal min 0.946liter 4quart 1min min (d) 1SCFM 1 ft m 60min min m3 12 ft hr hr

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45 Problem 1.35 Sometimes engineering equations are used in which units are present in an inconsistent manner. For example, a parameter that is often used in describing pump performance is the specific speed, NScu, given by 1 2 N( rpm) Q( gpm) N Scu 3 4 Hft ( ) What are the units of specific speed? A particular pump has a specific speed of What will be the specific speed in SI units (angular velocity in rad/s)? Solution Using data from tables (e.g. Table G.2) N Scu 1 2 rpmgpm ft 1 2 rpmgpm 3 4 ft 2πrad 1rev 1min.. 60s 4quart 1gal m 3 1quart 1min 60s ft rad m 3 s s m 3 4 m 1 2

46 Problem 1.36 A particular pump has an engineering equation form of the performance characteristic equatio given by H (ft) x 10-5 [Q (gpm)] 2, relating the head H and flow rate Q. What are the units of the coefficients 1.5 and 4.5 x 10-5? Derive an SI version of this equation. Solution Dimensions of "1.5" are ft. Dimensions of "4.5 x 10-5 " are ft/gpm 2. Using data from tables (e.g. Table G.2), the SI versions of these coefficients can be obtained 1.5ft m 1.5ft 0.457m 1 12 ft ft gpm ft gpm m 1 12 ft 1gal 1quart 60s 4quart m 3 1min ft gpm m m 3 s 2 The equation is 2 Hm ( ) Q m3 s

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