Units and Dimensions Mars Climate Orbiter 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.

Quantity = numerical value & units Example 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 Units and Dimensions

What are the dimensions & (SI units) on the following ? velocity = acceleration = force = pressure = energy = L/t (m/sec) L/t 2 (m/sec 2 ) M*L/t 2 (Kg m/ sec 2 ) M/L*t 2 (Kg m / m 2 sec 2 ) M*L 2 /t 2 Kg m 2 / sec 2 )

Typically, coefficients in physical laws (eg, KE = ½ mv 2 ),exponents, and arguments (log x, sin x, exp x = ex) have no dimensions. There are special dimensionless numbers used in chemical engineering; for example: Prandtl Number Reynolds Number

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) * ft / m & t(s) = t’(min) 60 s / min Thus, D’(m) = 3*[60 t’(min)] + 4 D’(m) = 55t’(min) 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

10 Minute Problem An empirical equation for calculating the inside heat transfer coefficient, h i, for the turbulent flow of liquids in a pipe is given by: where h i = 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 h i 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 lb f ) = mass * acceleration (F = m *a) Mass = kg (SI), g (CGS), or lb m (English)

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 X 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) = 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 lb mol AgNO 3. b g mol AgNO 3 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 CaCO 3. b. 150 kg of CaCO 3. c. 100 lb of CaCO 3.

Density, Specific Gravity, API Gravity Density =  [=] M/L 3 → kg/m 3, lb m / ft 3, g/cc, etc.  ≠ constant → f(T,P) Specific volume = V = volume / unit mass =  –1 [=] L 3 /M Specific gravity = sp gr = SG = For liquids & solids:  ref = H 2 O (liquid) at 4°C & 1 atm [  water = 1 g/cm 3 = 1000 kg/m 3 = lbm/ft 3 ] For gases:  ref = air at “standard conditions”

Tabulated Specific Gravities Example: SG of Ethanol at 140 F

API Gravity (Crude Oil)

Example The density of a liquid is 1500 kg/m 3 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 (ft 3 ) does 140 lb m of this material occupy at 20°C ?

Composition Mole fraction = Mass fraction = Volume fraction (gas) ????

Example A liquefied mixture of n-butane, n-pentane and n-hexane has the following composition in weight percent. n - C 4 H 10 = 50 % n - C 5 H 12 = 30 % n - C 6 H 14 = 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 ? CO CO6.0 CH H N Total 100.0

Concentration Concentration = quantity of A / volume kg / m 3 kg mol / m 3 g/L g /cc lb / ft 3 lb mol / ft 3

Example A solution of HNO 3 in water has a specific gravity of 1.10 at 25 C. The concentration of HNO 3 is 15 g/L. What is the mole fraction of HNO 3 in the solution ? What is the ppm (wt) of HNO 3 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 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/cm 3

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 ) T (°R) = T (°F ) T (°R) = T (°K ) * 1.8

Example  T ≠ T - conversions approaches are different Given the following equation: Where:  [=] gm / cm 3, 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:  [=] lb m / ft 3, 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:

Direction of fluid pressure on boundaries Furnace ductPipe or tube Heat exchanger Dam Pressure is a Normal Force (acts perpendicular to surfaces) It is also called a Surface Force

Units for Pressure UnitDefinition or Relationship 1 pascal (Pa)1 kg m -1 s -2 1 bar1 x 10 5 Pa 1 atmosphere (atm)101,325 Pa 1 torr1 / 760 atm 760 mm Hg1 atm pounds per sq. in. (psi) 1 atm

Standard Atmosphere 1 Atmosphere ft of water (ft H 2 0) psi (lb f / in 2 ) in Hg 760 mm Hg X 10 5 Pascal (Pa) 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: For an infinitesimal element (  z  0)

Incompressible fluid Liquids are incompressible i.e. their density is assumed to be constant: By using gauge pressures we can simply write: P o is the pressure at the free surface (P o =P atm ) When we have a liquid with a free surface the pressure P at any depth below the free surface is:

Measurement of Pressure Differences Apply the basic equation of static fluids to both legs of manometer, realizing that P 2 =P 3.

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 The manometer liquid is mercury, with a specific gravity (60°/60°) of 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/cm 3. The density of mercury is g/cm 3 The pressure gauge (PG) reads 33.1 psig. What is the pressure in kPa of the gas ? PG Gas 20 in24 in 3 in 16 in 3 in 12 in

Extra Practice Problems Problem Set Handout: I-52 – I-63