AE 205 Materials and Energy Balances Asst. Prof. Dr. Tippabust Eksangsri Chapter 2 Dimensions, Units, and Unit Conversion
Dimensions Dimensions are concepts of measurement in engineering works. The basic dimensions we are familiar with are length, mass, temperature and time. Other dimensions are called derived dimensions, because they are derived from the basic dimensions. The common derived dimensions are force, pressure, energy, concentration, etc.
Relation between basic and derived dimensions Density Volume Volume Flow Rate Area Time Mass Flow Rate Mass Length Velocity Acceleration Force
Units Units are the means of expressing the dimensions such as metre(m) for length, kilogram(kg) for mass, degree Celcius( C) for temperature and second(s) for time. Derived units are those that can be developed in terms of fundamental units such as Newton(N) for force, Pascal(Pa) for pressure, Joules(J) for energy and Molar(M) for concentration.
Common Dimensions and Units (SI) Dimensions Units Symbols for units Length metre m Mass kilogram kg Time second s Temperature Kelvin K Force Newton N Molar amount mole mol Energy Joule J Power Watt W Density kilogram per cubic metre Kg/m 3 Velocity metre per second m/s Acceleration metre per second squared m/s 2 Pressure Pascal Pa Heat Capacity Joule per kilogram Kelvin J/kg K
Common Dimensions and Units (AE) Dimensions Units Symbols for units Length foot ft Mass pound mass lb m Time second, minute, hour, day s, min, hr, day Temperature degree Rankine or degree Fharenheit R or F Force pound force lb f Molar amount pound mole lb mol Energy British thermal unit Btu Power horsepower hp Density pound mass per cubic foot lb m /ft 3 Velocity feet per second ft/s Acceleration feet per second squared ft/s 2 Pressure pound force per square inch psi Heat Capacity Btu per pound mass per degree F Btu/lb m F
Thermodynamics Temperature Scales S.I. System; Kelvin scale (K) T (K) = T (C) + 273.15 T (K) = T (C) British System or American Engineering System; Rankin scale (R) T (R) = T (F) + 459.67 T (R) = T (F)
Examples of Derived Dimension 1. Force (F) In English system, 1 lb f is a force required to accelerate a mass of 32.174 lb m at a rate of 1 ft/s 2 or 1 lbf = 32.174 lbm ft/s 2 In SI, 1 N is a force required to accelerate a mass of 1 kg at a rate of 1 m/s 2 or 1 N = 1 kg m/s 2
Example 1: Unit of Force From the definition F = ma When F = force, m = mass, and a = acceleration Then, F = m(kg) x a(m/s 2 ) F(kg m/s 2 ) While the definition of 1N is the movement of 1 kg-mass with the acceleration of 1 m/s 2 Hence, 1 N = 1 kg m/s 2
Examples of Derived Dimension 2. Pressure (P) Pressure is a force exerted by fluid per unit area Or P = F/A SI; Unit of pressure is Pascal (1 Pa =N/m 2 ) English; Unit of pressure is psi, (1 psi = 1lb f /in 2 )
Example 2: Unit of Pressure From the definition P = F/A When P = pressure, F = force, and A= cross-sectional area Therefore, P = F(N)/A(m 2 )= F(kg m/s 2 )/(A (m 2 ) P (kg/m s 2 ) Or P (Pascal) since 1 Pa = 1 kg/m s 2 Pascal is usually used as a common unit for pressure.
Units Conversion Conversion of units using conversion factors Conversion of units using prefix factors
Conversion for Fundamental Units Mass (m) 1 lb m = 0.45359 kg Length (s) 1 ft = 0.3048 m Temperature (T) T( F) = 1.8 T( C) + 32 T(R) = 1.8T(K)
Common Conversion Factors Dimensions Length Mass Time Temperature Force Molar amount Energy Power Pressure Conversion Factors 1 m = 39.370 in = 3.2808 ft 1 kg = 2.2046226 lb m Same unit T(K) = T( C) + 273.15, T(R) = 1.8 T(K) T ( F) = 1.8 T( C) + 32 1 N = 0.22481 lb f 1 lb f = 32.174 lb m ft/s 2 1 mol = 1 gmol = 0.0022 lb mol 1 kj = 0.94728 Btu 1W = 1 J/s 1 kw = 1.341 hp 1 Pa = 1N/m 2 = 1.4504 x 10-4 psi 1 atm = 101.325 kpa = 1.01325 bars = 760 mm.hg
Example 3A: Convert 2 km to miles. Conversion Factor is 1 mile = 1.61 km Example 3B: Convert 400 in 3 /day to cm 3 /min Conversion Factor is 1 inch = 2.54 cm 1 day = 24 hours 1 hour = 60 minutes
Example 4: Unit conversion for biochemical engineering Glucoamylase is an enzyme that aids in the conversion of starch to glucose. Experiments show that 1 g mol of glucoamylase in a 4% starch solution results in a production rate of glucose of 0.6 g mol/ml min. Determine the production rate of glucose for this system in the units of lb mol/ft 3 day.
Standard prefixes in SI units Multiple Prefix Multiple Prefix 10 12 Tera (T) 10 9 Giga (G) 10 6 Mega (M) 10 3 Kilo (k) 10 2 Hecto (h) 10 1 Deka (da) 10-12 Pico (p) 10-9 Nano (n) 10-6 Micro ( ) 10-3 Milli (m) 10-2 Centi (c) 10-1 Deci (d)
Example 5: Conversion of particle size A semiconductor is ZnS with a particle diameter of 1.8 nm. 5A: Convert this value to dm 5B: Convert this value to inches
Dimension Consistency (Homogeneity) Basic principle states that equations must be dimensionally consistent. Each term in an equation must have the same net dimensions and units as every other term to which it is added, subtracted or equated. Dimensional considerations can be used to help identify the dimensions and units of or quantities in an equation.
Example 6: Conversion of particle size The handbook shows that microchip etching roughly follows the relation; d = 16.2 16.2e -0.021t Where d = depth of the etch in microns t = time of the etch in seconds (A) What are the unit associated with the numbers 16.2 and 0.021? (B) Convert the relation so that d is expressed in inches and t is in minutes.