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Units and Dimensions SEPTEMBER 30, 1999 Likely Cause Of Orbiter Loss Found The peer review preliminary findings indicate that one team used English units (e.g., inches, feet and pounds) while the other used metric units for a key spacecraft operation. Mars Climate Orbiter Units and Dimensions Quantity = numerical value & units Example - 100 kg/hr Dimensions = basic concepts of measurement Units = quantitatively expressing dimensions All dimensions of interest can be expressed in terms of: Mass Length Temperature Time Currency What are the dimensions & (SI units) on the following ? velocity = acceleration = force = pressure = energy = L/t (m/sec) L/t2 (m/sec2 ) M*L/t2 (Kg m/ sec2) M/L*t2 (Kg m / m2 sec2) M*L2/t2 Kg m2 / sec2) Typically, coefficients in physical laws (eg, KE = ½ mv2),exponents, and arguments (log x, sin x, exp x = ex) have no dimensions. There are special dimensionless numbers used in chemical engineering; for example: Reynolds Number Prandtl Number N Re N Pr DV Cp k Dimensional Homogeneity & Dimensionless Numbers • every added and subtracted term in any equation must have the same dimensions. Multiplication & Division of quantities • creates compound dimensions and units Addition and Subtraction of quantities • must have same dimensions & units Example Consider the equation D(ft) = 3t(s) + 4 • What are the dimensions and units of 3 and 4 ? Convert the equation D(ft) = 3t(s) + 4 to D’(m) = __t’(min) + __ Convert each term then substitute ... D(ft) = D’(m) * 3.2808 ft / m & t(s) = t’(min) 60 s / min Thus, 3.2808D’(m) = 3*[60 t’(min)] + 4 D’(m) = 55t’(min) + 1.22 • What are the dimensions & units of 55 and 1.22 ? Example You are traveling at 51 km/hr and increase your speed by 1 ft/s; what is your new velocity? Can you add these because they have the same dimensions ? Dimensional ledger/ equations • think units first, then numerical values • break big problem down km km ft ?? V 51 1 * hr hr sec ?? 10 Minute Problem An empirical equation for calculating the inside heat transfer coefficient, hi, for the turbulent flow of liquids in a pipe is given by: 0.023 G 0.8 K 0.67Cp0.33 hi D0.2 0.47 where hi = heat transfer coefficient, Btu/(hr)(ft)2(°F) G = mass velocity of the liquid, lbm/(hr)(ft)2 K = thermal conductivity of the liquid, Btu/(hr)(ft)(°F) Cp = heat capacity of the liquid, Btu/(lbm)(°F) μ = Viscosity of the liquid, lbm/(ft)(hr) D = inside diameter of the pipe, (ft) a. Verify if the equation is dimensionally consistent. b. What will be the value of the constant, given as 0.023, if all the variables in the equation are inserted in SI units and hi is in SI units. Extra Practice Problems Problem Set Handout: I-1 – I-17 Mass, Weight, and Force Mass: amount of material - mass ≠ weight Weight: Force that material exerts due to gravity (g) which changes with location, etc. Force: (Newton, dyne, or lbf) = mass * acceleration (F = m *a) Mass = kg (SI), g (CGS), or lbm (English) kg m F (N ) 2 s lbm ft F (lb f ) C * 2 s 32.174 lbm ft / sec 2 gc lb f 10 Minute Problem Momentum (lbf) is equal to mass (lbm / sec) X velocity (ft/sec) Determine the momentum force transferred to a wall by a stream of water flowing from a fire hose at 50 ft/sec and 1000 lb/hr. Extra Practice Problems Problem Set Handout: I-18 – I-21 Moles, Density and Concentration Moles Mole = certain number of entities 6.023 X 1023 molecules • g-mole = amt of substance whose mass in grams is equal to the molecular weight of the substance • similarly kg-mole & lb-mole • molecular weight (MW) = mass mole • atomic weight - atomic mass .... Inside back cover of textbook 10 Minute Problem Silver nitrate (lunar caustic) is a white crystalline salt, used in marking inks, medicine and chemical analysis. How many kilograms of silver nitrate (AgNO3) are there in : a. 13.0 lb mol AgNO3. b. 55.0 g mol AgNO3 Calcium carbonate is a naturally occurring white solid used in the manufacture of lime and cement. Calculate the number of lb mols of calcium carbonate in: a. 50 g mol of CaCO3. b. 150 kg of CaCO3. c. 100 lb of CaCO3. Density, Specific Gravity, API Gravity Density = [=] M/L3 → kg/m3, lbm / ft3, g/cc, etc. • ≠ constant → f(T,P) Specific volume = V = volume / unit mass = –1 [=] L3/M Specific gravity = sp gr = SG • For liquids & solids: = A (T , P) ref (Tref , Pref ) ref = H2O(liquid) at 4°C & 1 atm [water= 1 g/cm3 = 1000 kg/m3 = 62.43 lbm/ft3] • For gases: ref = air at “standard conditions” Tabulated Specific Gravities Example: SG of Ethanol at 140 F F 140 48.2 EtOH SG 40 F 0.772 H 20 62.4 API Gravity (Crude Oil) 141.5 API fluid, 60 F sp.gr. water , 60 F 131.5 Example The density of a liquid is 1500 kg/m3 at 20°C. • What is the specific gravity 20°C/4°C of this material ? • What is the API gravity of the liquid ? • What volume (ft3) does 140 lbm of this material occupy at 20°C ? Composition • Mole fraction = moles (n) of A total (nT ) moles mass (m) of A • Mass fraction = total ( mT ) mass • Volume fraction (gas) ???? n RT V P Example A liquefied mixture of n-butane, n-pentane and n-hexane has the following composition in weight percent. n - C4H10 = 50 % n - C5H12 = 30 % n - C6H14 = 20 % Calculate the weight fraction, mol fraction and mol percent of each component and also the average molecular weight of the mixture. 10 Minute Problem A mixture of gases is analyzed and found to have the following composition (volume percent). How much will 3 lb mol of this gas weigh ? CO2 CO CH4 H2 N2 12.0 6.0 27.3 9.9 44.8 Total 100.0 Concentration Concentration = quantity of A / volume kg / m3 kg mol / m3 lb / ft3 g/L lb mol / ft3 g /cc Example A solution of HNO3 in water has a specific gravity of 1.10 at 25 C. The concentration of HNO3 is 15 g/L. What is the mole fraction of HNO3 in the solution ? What is the ppm (wt) of HNO3 in the solution ? 10 Minute Problem The 1993 Environmental Protection Agency (EPA) regulation contains standards for 84 chemicals and minerals in drinking water. According to the EPA one of the most prevalent of the listed contaminants is naturally occurring antimony. The maximum contaminant level for antimony and nickel has been set at 0.006 mg/L and 0.1 mg/L respectively. A laboratory analysis of your household drinking water shows the antimony concentration to be 4 ppb (wt) (parts per billion) and that of nickel to be 60 ppb (wt). Determine if the drinking water is safe with respect to the antimony and nickel levels. Assume density of water to be 1.00 g/cm3 Extra Practice Problems Problem Set Handout: I-22 – I-44 Temperature Temperature - average kinetic energy of molecules. Relative Fahrenheit (°F) Celsius (°C) Absolute Rankin ( °R ) Kelvin (°K) Conversions T (°K) = T (°C ) + 273.15 T (°R) = T (°F ) + 459.67 T (°R) = T (°K ) * 1.8 Example DT ≠ T - conversions approaches are different Given the following equation: (1.096 0.00086 T )e0.000953P Where: [=] gm / cm3, T [=] °C, P [=] atm A) Determine the units on the three constants B) Convert the constants to accurately reflect the following revised set of units: [=] lbm / ft3, T [=] °R, P [=] psi Extra Practice Problems Problem Set Handout: I-45 – I-51 Pressure Pressure is defined as the amount of force exerted on a unit area of a substance: force N P 2 Pa area m force lb f P 2 area ft Direction of fluid pressure on boundaries Furnace duct Pipe or tube Heat exchanger Pressure is a Normal Force (acts perpendicular to surfaces) It is also called a Surface Force Dam Units for Pressure Unit 1 pascal (Pa) Definition or Relationship 1 kg m-1 s-2 1 bar 1 x 105 Pa 1 atmosphere (atm) 101,325 Pa 1 torr 1 / 760 atm 760 mm Hg 1 atm 14.696 pounds per sq. in. (psi) 1 atm Standard Atmosphere 1 Atmosphere 33.91 ft of water (ft H20) 14.696 psi (lbf / in2) 29.92 in Hg 760 mm Hg 1.013 X 105 Pascal (Pa) 101.3 kPa Pressure distribution for a fluid at rest Let’s determine the pressure distribution in a fluid at rest in which the only body force acting is due to gravity The sum of the forces acting on the fluid must equal zero Pressure distribution for a fluid at rest A force balance in the z direction gives: F z 0 PS z PS z Dz SDzg Pz Dz Pz g Dz For an infinitesimal element (Dz0) dP g dz Incompressible fluid Liquids are incompressible i.e. their density is assumed to be constant: P2 P1 g ( z2 z1 ) When we have a liquid with a free surface the pressure P at any depth below the free surface is: P g h Po Po is the pressure at the free surface (Po=Patm) By using gauge pressures we can simply write: P gh Measurement of Pressure Differences Apply the basic equation of static fluids to both legs of manometer, realizing that P2=P3. P2 Pa b g ( Z m Rm ) P3 Pb b g ( Z m ) a gRm Pa Pb g Rm ( a b ) Example A U-tube manometer is used to determine the pressure drop across an orifice meter. The liquid flowing in the pipe line is a sulfuric acid solution having a specific gravity (60°/60°) of 1.250. The manometer liquid is mercury, with a specific gravity (60°/60°) of 13.56. The manometer reading is 5.35 inches, and all parts of the system are at a temperature of 60°F. What is the pressure drop across the orifice meter in psi ? 10 Minute Problem The barometric pressure is 720 mm Hg. The density of the oil is 0.80 g/cm3 . The density of mercury is 13.56 g/cm3 The pressure gauge (PG) reads 33.1 psig. What is the pressure in kPa of the gas ? 3 in Gas 12 in 20 in 24 in 3 in PG 16 in Extra Practice Problems Problem Set Handout: I-52 – I-63