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Reading Materials: Chapter 4

Reading Materials: Chapter 4. Describing Physical Quantities. LECTURES 4-5. Class Activity. 1. How tall are you? 2. How long does it take you to get to Adelaide Uni every morning? 3. How many days are there in one year? 4. How heavy are you?

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Reading Materials: Chapter 4

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  1. Reading Materials: Chapter 4 Describing Physical Quantities LECTURES 4-5 CHEM ENG 1007

  2. Class Activity 1. How tall are you? 2. How long does it take you to get to Adelaide Uni every morning? 3. How many days are there in one year? 4. How heavy are you? 5. What is the distance from the earth to the moon? 6. What is the average weight of the new-born baby? CHEM ENG 1007

  3. Objective • At the end of these lectures you should be able to: • convert between sets of units by using dimensional equation • add, subtract, multiply and divide units • use SI, American Engineering and cgs units • understand dimensional homogeneity CHEM ENG 1007

  4. §4.1 Units & Dimensions Quick Quiz 1: True or False A quantity is meaningless without units! "the distance from my house to school is six" unless we follow that statement with "miles" or "kilometres", or whichever unit, the above statement is meaningless. CHEM ENG 1007

  5. §4.1 Units & Dimensions Quick Quiz 2: True or False All additive/subtractive terms must have the same dimension! CHEM ENG 1007

  6. Illustration 1 Every Monday you buy three litres of milk and five packages of Tim Tam. Since you have to carry your purchases home, you’d like to know how many kilograms of food you’ve bought. Can we just add together litres of milk plus packages of tim tam to get kilograms of food? (3 L milk) + (5 packages of tim tam) = ? kg of food CHEM ENG 1007

  7. Illustration 2 Assignment 1, Question 7 6.201 cm + 7.4 cm + 0.68 cm + 12.0 cm = ? 1.6 km + 1.62 m + 1200 cm = ? 8.264 g - 7.8 g = ? 10.4168 m - 6.0 m = ? 12.00 m + 15.001 kg = ? 131 cm x 2.3 cm = ? 5.7621 m x 6.201 m = ? 20.2 cm divided by 7.41 s = ? 40.002 g divided by 13.000005 ml = ? CHEM ENG 1007

  8. Case Study 1: Feet or Meters? SEOUL, South Korea – A mix up in the cockpit over whether altitude guidance was measured in feet or meters led to the crash of a Korean Air Lines McDonnell Douglas MD-11 freighter soon after takeoff in Shanghai in April 1999. The crash killed all 3 crew-members. Five people on the ground were killed and 40 more were injured when the plane went down in light rain onto a construction site near Shanghai’s Hongqiao Airport. CHEM ENG 1007

  9. Case Study 1: Feet or Meters? A Chinese air-traffic controller directed the pilots to an altitude of 1,500 meters (4,950 feet). The plane was climbing rapidly to that level when the co-pilot told the pilot he thought the instructed height was 1,500 feet (455 meters). The international aviation industry commonly measures altitude in feet, and the confusion led the pilot to conclude the jet was almost 1,000 meters too high, so he quickly moved the controls to lower the plane. As the plane descended, the pilot realized the error but couldn’t correct the mistake in time. South Korea’s Ministry of Construction and Transportation said Korean Air Lines would lose the right to serve the Seoul-Shanghai cargo route for at least two years because of errors by the pilots. Korean Air Lines said it would appeal the decision….. Wall Street Journal, June 6, 2001 CHEM ENG 1007

  10. Case Study 2 Mars Probe Lost Due to Simple Math Error NASA lost its $125-million Mars Climate Orbiter because spacecraft engineers failed to convert from English to metric measurements when exchanging vital data before the craft was launched. pound-seconds versus newton-seconds Want to know more….. http://articles.latimes.com/1999/oct/01/news/mn-17288 CHEM ENG 1007

  11. Dimensions versus Units Dimensions are properties that can bemeasured. Four basic dimensions:length [L], mass [M], time [T], and temperature [] Units are specific magnitudes of dimensions. Different units can measure the same dimension Example: Dimension: length Units: foot, metre, yard, mile, etc. Such units are related by a Conversion factor CHEM ENG 1007

  12. 0 0.454 kg 1 kg 2 kg  1 lbm 2 lbm Illustration 3 Dimension versus units • Dimension: mass • Units: kilogram (kg) and pound (lbm) Conversion factor: 1 lbm = 0.453593 kg 1 kg = 2.2046 lbm CHEM ENG 1007

  13. Systems of Units Three Base Unit groupings are: • Metric system (also known as SI system) • American / British engineering system. • CGS (Centimetres-Grams-Seconds) system. Should be Kelvin CHEM ENG 1007

  14. Concept Inventory • Which of the following set of units is in CGS system • kg, m, s, N • lbm, ft, s, lbf • g, m, s, N • g, cm, s, dyne • kg, cm, s, dyne CHEM ENG 1007

  15. §4.1.1 Conversion Factors A conversion factor is an equation that equates two quantities expressed in different units where, in fact, the two quantities are exactly equivalent. Examples: 12 in = 1 ft 1000 g = 1 kg 60s = 1 min CHEM ENG 1007

  16. Conversion Factors Table Or turn to page xi of the text-book CHEM ENG 1007

  17. §4.1.1 Conversion Factors • Conversion factors  dimensionless quantities. • Enable conversion between different units systems. • Conversion factors do not change magnitude, just value of units. CHEM ENG 1007

  18. Rearrange?: Dimensionless?:  Illustration 4 Is Conversion factor of metres to feet really dimensionless? CHEM ENG 1007

  19. Factor Prefix Symbol Factor Prefix Symbol 109 giga G 10-1 deci d 106 mega M 10-2 centi c 103 kilo k 10-3 milli m 102 hecto h 10-6 micro  101 deka da 10-9 nano n Prefixes Prefixes are used in SI to indicate powers of ten. Multiples or fractions of basic units defined for convenience. Commonly used for Joule, Pascal, Metre, Byte CHEM ENG 1007

  20. Quick Quiz How many Megabytes in 1Gigabytes? From the prefixes table, we get the following conversion factors 109 Bytes = 1 GB 106 Bytes = 1 MG By dimensional equation CHEM ENG 1007

  21. Quick Quiz How many micrograms in 5 kilograms? Again from the prefixes table, we get the following conversion factors 103 g = 1 kg 10-6 g = 1 g CHEM ENG 1007

  22. Quick Quiz How many millimetres in 1 nanometres? Again from the prefixes table, we get the following conversion factors 10-3 m = 1 mm 10-9 m = 1 nm CHEM ENG 1007

  23. Quick Quiz • Write conversion factors that relate to each of the following pairs of units: • Litres and mL • Hours and seconds • mg and kg • Nanometers and kilometers 1 L = 1000 mL 1 h = 3600 s 106 mg = 1 kg 1012 nm = 1 km CHEM ENG 1007

  24. Unit Conversion • Conversion of units from one system to another is necessary in process calculations and analysis, when • Data is from different sources, or • Variables are measured from instruments of different standards • (starting quantity) x (conversion factor) = equivalent quantity CHEM ENG 1007

  25. Illustration 5 Convert 28 inches to its equivalent number of feet. 12 in = 1 ft When solving a problem, the idea is to set up an equation so that all unwanted units cancel, leaving only the desired units. Convert mass of 60 lbm into kg?  1 lbm = 0.454 kg CHEM ENG 1007

  26. Dimensional Equation • A convenient method for unit conversion. • How to set up a dimensional equation? Write the given quantity and its units on the left, write the units of conversion factors that cancel the old units and replace them with the described ones, fill in the values of the conversion factors, and carry out the indicated arithmetic to find the desired value. CHEM ENG 1007

  27. Illustration 6 • Convert a mass of 150 lbm into equivalent in kg. • Solution: • If 1 lbm is equivalent to 0.454 kg then • or by dimensional equation CHEM ENG 1007

  28. Illustration 7 Using dimensional equation, convert a weight 90 N into lbf ? 1 N = 0.2281 lbf Using dimensional equation, convert 8 m/s into ft/min? 1 m = 3.2808 ft; 1 min = 60 s CHEM ENG 1007

  29. Quick Quiz • Convert a 40 mile/gal into km/L • Convert a force of 40 lbf into equivalent in N. CHEM ENG 1007

  30. §4.1.5 Combined Units Derived as combination of the base units: by multiplying or dividing of the base units velocity (m/s), area (m2), flow rate (ft3s-1) CHEM ENG 1007

  31. §4.1.5 Combined Units To produce conversion factors for combined units, conversion factors of base units can be raised to any power. Examples: CHEM ENG 1007

  32. Illustration 8 Convert 25 lbm.ft/min2 to its equivalent in kg.cm/s2? CHEM ENG 1007

  33. Quick Quiz Convert 45 m3/d2 to its equivalent in mm3/s2?   CHEM ENG 1007

  34. Coherent derived units Non-coherent derived units §4.1.5 Combined Units • Derived equivalentsof combined units: • 1 N = 1 kgm s-2 • 1 dyne = 1 gcm s-2 • 1 lbf = 32.174 lbmft s-2 • 1 Pa = 1 Nm s-2 • 1 gal = 0.16 ft3 (imperial) • 1 gal = 0.133 ft3 (US) CHEM ENG 1007

  35. Coherent versus Non-coherent derived Units Coherent Derived Units: Value of defined unit = Combined value of Basic Units e.g. 1 J (joule) = 1 (kg.m2)/s2 Non-coherent Value of defined unit  Value of base units e.g. 1 Btu  1 (kg.m2)/s3  Conversion factor required e.g. 9.486x10-4 Btu = 1 (kg.m2)/s3 Many American/British defined units are non-coherent (e.g. Btu, lbf) CHEM ENG 1007

  36. Summary for Units Conversion To convert a quantity expressed in terms of one unit to its equivalent in terms of another unit, multiply the given quantity by the conversion factor (new unit/old unit). Alternative way: CHEM ENG 1007

  37. §4.2.5 Dimensional Consistency • Rules of dimensional consistency 1. Terms that are added together (or subtracted) must have the same units. For example, in the equation Q = ab + c2, the units of ab must be the same as those of c2. 2. Exponents must be unitless (dimensionless). Thus, if an exponent consists of several terms, the units of all those terms must cancel. For example, in the equation y = xab/c, the units in the term ab/c must all cancel out to leave no units. CHEM ENG 1007

  38. Illustration 9 Arrhenius equation CHEM ENG 1007

  39. Illustration 10 Ideal Gas Equation In terms of these parameters, show that ideal gas equation obeys the laws of dimensional consistency CHEM ENG 1007

  40. Illustration 10 Substitute units into Equation? CHEM ENG 1007

  41. Illustration 10 Substitute base dimensions into Equation? CHEM ENG 1007

  42. Illustration 11 Consider this equation What are SI units of ? CHEM ENG 1007

  43. Illustration 11 Compare dimensional homogeneity! Rearrange yields units for . CHEM ENG 1007

  44. §4.1.6 Force • The force “F” to accelerate a mass “m” at an acceleration rate “a” is defined by Newton’s second law as • F = ma [M.L.T-2] • One form of acceleration is that associated with earth’s gravity. The rate of acceleration is described by the gravitational acceleration, g, and the force that an object exerts on the earth’s surface (its “weight”), is • W = mg CHEM ENG 1007

  45. §4.1.6 Force Factors for Unit conversions of Force 1 N = 1 kg.m/s2 = 105 dynes = 105 g.cm/s2 = 0.22481 lbf CHEM ENG 1007

  46. Example 4.2 An object has a mass equal to 1 lbm. What is its weight in pounds-force (lbf) Solution: From Table 4.3, we note that g = 32.174 ft/s2 CHEM ENG 1007

  47. Example 4.3 • An object has a mass equal to 8.41 kg. What is its weight • in Newtons (N) • pounds-force (lbf) • Solution: CHEM ENG 1007

  48. Quick Quiz From Reading Question 4: The weight of an astronaut is measured on a distant planet and found to be one-fifth of his weight on the earth’s surface. Is his mass different on that distant planet than on earth? What does the weight difference imply about the acceleration of gravity on the distant planet? CHEM ENG 1007

  49. Summary • 2 case-studies • Identified the difference in SI, American Engineering, and CGS units system • Understand • conversion factors • dimensional equations • dimensionless • dimensional consistency • coherent and non-coherent derived units • Force/Weight • Able to convert between sets of units CHEM ENG 1007

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