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- 1. 2003 by CRC Press LLC 11 2 Unit Systems: Dimensional Analysis and Similarity 2.1 Magnitude and Unit Systems The value of any physical magnitude is expressed as the product of two factors: the value of the unit and the number of units. The physical properties of a system are related by a series of physical and mechanical laws. Some magnitudes may be considered fundamental and others derived. Fundamen- tal magnitudes vary from one system to another. Generally, time and length are taken as fundamental. The unit systems need a third fundamental magnitude, which may be mass or force. Those unit systems that have mass as the third fundamental magnitude are known as absolute unit systems, while those that have force as a fundamental unit are called technical unit systems. There are also engineering unit systems that consider length, time, mass, and force as fundamental magnitudes. 2.1.1 Absolute Unit Systems There are three absolute unit systems: the c.g.s. (CGS), the Giorgi (MKS), and the English (FPS). In all of these, the fundamental magnitudes are length, mass, and time. The different units for these three systems are shown in Table 2.1. In these systems, force is a derived unit dened beginning with the three fundamental units. The force and energy units are detailed in Table 2.2. When heat magnitudes are used, it is convenient to dene the temperature unit. For the CGS and MKS systems, the unit of temperature is degrees Centigrade (C), while for the English system it is degrees Fahrenheit (F). Heat units are dened independently of work units. Later, it will be shown that relating work and heat requires a factor called the mechanical equivalent of heat. TX69299 ch01 frame.book Page 11 Wednesday, September 4, 2002 2:13 PM
- 2. 2003 by CRC Press LLC 12 Unit Operations in Food Engineering 2.1.2 Technical Unit Systems Among the most used technical systems are the metric and the English systems. In both, the fundamental magnitudes are length, force, and time. In regard to temperature, the unit of the metric system is the Centigrade degree, and that of the English system is the Fahrenheit. Table 2.3 shows the fundamental units of the metric and English systems. In engineering systems, mass is a derived magnitude, which in the metric system is 1 TMU (technical mass unit) and in the English system is 1 slug. 2.1.3 Engineering Unit Systems Until now, only unit systems that consider three magnitudes as fundamental have been described. However, in engineering systems, four magnitudes are considered basic: length, time, mass, and force. Table 2.4 presents the differ- ent units for the metric and English engineering systems. TABLE 2.1 Absolute Unit Systems Magnitude System c.g.s. Giorgi English (CGS) (MKS) (FPS) Length (L) 1 centimeter (cm) 1 meter (m) 1 foot (ft) Mass (M) 1 gram (g) 1 kilogram (kg) 1 pound-mass (lb) Time (T) 1 second (s) 1 second (s) 1 second(s) TABLE 2.2 Units Derived from Absolute Systems Magnitude System c.g.s. Giorgi English (CGS) (MKS) (FPS) Force 1 dyne 1 Newton (N) 1 poundal Energy 1 erg 1 Joule (J) 1 (pound)(foot) TABLE 2.3 Technical Unit Systems Magnitude System Metric English Length (L) 1 meter (m) 1 foot (ft) Force (F) 1 kilogram force (kp or kgf) 1 pound force (lbf) Time (T) 1 second (s) 1 second (s) Temperature () 1 degree Centigrade (C) 1 degree Fahrenheit (F) TX69299 ch01 frame.book Page 12 Wednesday, September 4, 2002 2:13 PM
- 3. 2003 by CRC Press LLC Unit Systems: Dimensional Analysis and Similarity 13 When dening mass and force as fundamental, an incongruity may arise, since these magnitudes are related by the dynamics basic principle. To avoid this incompatibility, a correction or proportionality factor (gc) should be inserted. The equation of this principle would be: Observe that gc has mass units (acceleration/force). The value of this correction factor in the engineering systems is: 2.1.4 International Unit System (IS) It was convenient to unify the use of the unit systems when the AngloSaxon countries incorporated the metric decimal system. With that purpose, the MKS was adopted as the international system and denoted as IS. Although the obligatory nature of the system is recognized, other systems are still used; however, at present many engineering journals and books are edited only in IS, making it more and more acceptable than other unit systems. Table 2.5 presents the fundamental units of this system along with some supplemen- tary and derived units. Sometimes the magnitude of a selected unit is too big or too small, making it necessary to adopt prexes to indicate multiples and submultiples of the fundamental units. Generally, it is advisable to use these multiples and TABLE 2.4 Engineering Unit Systems Magnitude System Metric English Length (L) 1 meter (m) 1 foot (ft) Mass (M) 1 kilogram (kg) 1 pound-mass (lb) Force (F) 1 kilogram force (kp or kgf) 1 pound force (lbf) Time (T) 1 second (s) 1 second (s) Temperature () 1 degree Centigrade (C) 1 degree Fahrenheit (F) gc Force = Mass Acceleration Metric system: 9.81 kgmass meter kgforce second 9.81 kg m kg s2 2 gC = ( )( ) ( )( ) = English system: 32.17 lbmass foot lbforce second 32.17 lb ft lbf s2 2 gC = ( )( ) ( )( ) = TX69299 ch01 frame.book Page 13 Wednesday, September 4, 2002 2:13 PM
- 4. 2003 by CRC Press LLC 14 Unit Operations in Food Engineering submultiples as powers of 103. Following is a list of the multiples and sub- multiples most often used, as well as the name and symbol of each. It is interesting that, in many problems, concentration is expressed by using molar units. The molar unit most frequently used is the mole, dened as the quantity of substance whose mass in grams is numerically equal to its molec- ular weight. 2.1.5 Thermal Units Heat is a form of energy; in this way, the dimension of both is ML2T2. However, in some systems temperature is taken as dimension. In such cases, heat energy can be expressed as proportional to the product mass times temperature. The proportionality constant is the specic heat, which depends on the material and varies from one to another. The amount of heat is dened as a function of the material, with water taken as a reference and the specic heat being the unit, so: TABLE 2.5 International Unit System Magnitude Unit Abbreviation Dimension Length meter m L Mass kilogram kg M Time second s T Force Newton N MLT2 Energy Joule J ML2T2 Power Watt W ML2T3 Pressure Pascal Pa ML1T2 Frequency Hertz Hz T1 Prex Multiplication Factor IS Symbol tera 1012 T giga 109 G mega 106 M kilo 103 k hecto 102 h deca 101 da deci 101 d centi 102 c mili 103 m micro 106 nano 109 n pico 1012 p femto 1015 f atto 1018 a Heat Mass Specific heat Temperature= TX69299 ch01 frame.book Page 14 Wednesday, September 4, 2002 2:13 PM
- 5. 2003 by CRC Press LLC Unit Systems: Dimensional Analysis and Similarity 15 The heat unit depends on the unit system adopted. Thus: Metric system: Calorie: heat needed to raise the temperature of a gram of water from 14.5 to 15.5C English systems: Btu (British thermal unit): quantity of heat needed to raise the temperature of a pound of water one Fahrenheit degree (from 60 to 61F) Chu (Centigrade heat unit or pound calorie): quantity of heat needed to raise the temperature of one pound of water one degree Centigrade International system: Calorie: since heat is a form of energy, its unit is the Joule. The calorie can be dened as a function of the Joule: 1 calorie = 4.185 Joules Since heat and work are two forms of energy, it is necessary to dene a factor that relates them. For this reason, the denominated mechanical equiv- alent of heat (Q) is dened so that: so: 2.1.6 Unit Conversion The conversion of units from one system to another is easily carried out if the quantities are expressed as a function of the fundamental units mass, length, time, and temperature. The so-called conversion factors are used to convert the different units. The conversion factor is the number of units of a certain system contained in one unit of the corresponding magnitude of another system. The most common conversion factors for the different mag- nitudes are given in Table 2.6. When converting units, it is necessary to distinguish the cases in which only numerical values are converted from those in which a formula should be converted. When it is necessary to convert numerical values from one unit to another, the equivalencies between them, given by the conversion factors, are used directly. Q =Heat energy Mechanical energy Q = = = Mechanical energy Heat energy MLT L M L T 2 2 2 1 TX69299 ch01 frame.book Page 15 Wednesday, September 4, 2002 2:13 PM
- 6. 2003 by CRC Press LLC 16 Unit Operations in Food Engineering Table 2.6 Conversion Factors Mass 1 lb 0.4536 kg (1/32.2) slug Length 1 inch 2.54 cm 1 foot 0.3048 m 1 mile 1609 m Surface 1 square inch 645.2 mm2 1 square foot 0.09290 m2 Volume and Capacity 1 cubic foot 0.02832 m3 1 gallon (imperial) 4.546 l 1 gallon (USA) 3.786 l 1 barrel 159.241 l Time 1 min 60 s 1 h 3600 s 1 day 86,400 s Temperature difference 1C = 1 K 1.8F Force 1 poundal (pdl) 0.138 N 1 lbf 4.44 N 4.44 105 dina 32.2 pdl 1 dyne 105 N Pressure 1 technical atmosphere (at) 1 kgf/cm2 14.22 psi 1 bar 100 kPa 1 mm Hg (tor) 133 Pa 13.59 kgf/cm2 1 psi (lb/in2) 703 kgf/m2 Energy, Heat, and Power 1 kilocalorie (kcal) 4185 J 426.7 kgfm TX69299 ch01 frame.book Page 16 Wednesday, September 4, 2002 2:13 PM
- 7. 2003 by CRC Press LLC Unit Systems: Dimensional Analysis and Similarity 17 In cases of conversion of units of a formula, the constants that appear in the formula usually have dimensions. To apply the formula in units different from those given, only the constant of the formula should be converted. In cases in which the constant is dimensionless, the formula can be directly applied using any unit system. 2.2 Dimensional Analysis The application of equations deducted from physical laws is one method of sol