units and dimensions mars climate orbiter september 30, 1999 likely cause of orbiter loss found the...
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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.
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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:
MassLength
TemperatureTime
Currency
Units and Dimensions
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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)
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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 chemicalengineering; for example:
Prandtl Number
Reynolds NumberDV
N Re
k
CN pPr
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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
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ExampleConsider 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 ?
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Example
You are traveling at 51 km/hr and increase your speedby 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
??
??*
sec151
ft
hr
km
hr
kmV
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10 Minute ProblemAn empirical equation for calculating the inside heat transfer coefficient, h i, for the turbulent flow of liquids in a pipe is given by:
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.
47.02.0
33.067.08.0023.0
D
CpKGhi
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Extra Practice Problems
Problem Set Handout: I-1 – I-17
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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)
2)(
s
mkgNF
2*)(
s
ftlbClbF m
f f
mc lb
ftlbg
2sec/174.32
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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.
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Extra Practice Problems
Problem Set Handout: I-18 – I-21
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Moles, Density and Concentration
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Moles
Mole = certain number of entities
6.023 X 1023 molecules
• g-mole = amt of substance whose mass in grams isequal to the molecular weight of the substance
• similarly kg-mole & lb-mole
• molecular weight (MW) =
• atomic weight - atomic mass .... Inside back cover of textbook
mole
mass
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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.
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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: 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”
),(
),(
refrefref
A
PT
PT
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Tabulated Specific Gravities
Example: SG of Ethanol at 140 F
772.04.62
2.484020
140
F
H
FEtOHSG
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API Gravity (Crude Oil)
5.131
..
5.141
60,
60,
Fwater
Ffluidgrsp
API
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Example
The density of a liquid is 1500 kg/m3 at 20°C.
• What is the specific gravity 20°C/4°C of thismaterial ?
• What is the API gravity of the liquid ?
• What volume (ft3) does 140 lbm of this materialoccupy at 20°C ?
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Composition
• Mole fraction =
• Mass fraction =
• Volume fraction (gas) ????
molesntotal
Aofnmoles
T )(
)(
massmtotal
Aofmmass
T )(
)(
P
TRnV
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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.
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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 12.0CO 6.0CH4 27.3H2 9.9N2 44.8
Total 100.0
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Concentration
Concentration = quantity of A / volume
kg / m3 kg mol / m3 g/L g /cc
lb / ft3 lb mol / ft3
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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 ?
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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 antimonyconcentration 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
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Extra Practice Problems
Problem Set Handout: I-22 – I-44
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Temperature
Temperature - average kinetic energy of molecules.
Relative Fahrenheit (°F)Celsius (°C)
AbsoluteRankin ( °R ) Kelvin (°K)
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Conversions
T (°K) = T (°C ) + 273.15
T (°R) = T (°F ) + 459.67
T (°R) = T (°K ) * 1.8
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Example
T ≠ T - conversions approaches are different
Given the following equation:PeT 000953.0)00086.0096.1(
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
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Extra Practice Problems
Problem Set Handout: I-45 – I-51
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Pressure Pressure is defined as the amount of force exerted on a unit area of a substance:
2
2
ft
lb
area
forceP
Pam
N
area
forceP
f
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Direction of fluid pressure on boundaries
Furnace duct Pipe or tube
Heat exchanger
Dam
Pressure is a Normal Force(acts perpendicular to surfaces)It is also called a Surface Force
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Units for Pressure
Unit Definition or Relationship
1 pascal (Pa) 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
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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
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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
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Pressure distribution for a fluid at rest
zgSPSPSFzzzz
0
gz
PP zzz
A force balance in the z direction gives:
For an infinitesimal element (z0)
gdz
dP
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Incompressible fluidLiquids are incompressible i.e. their density is assumed to be constant:
By using gauge pressures we can simply write:
Po is the pressure at the free surface (Po=Patm)
When we have a liquid with a free surface the pressure P at any depth below the free surface is:
)( 1212 zzgPP
oPhgP
hgP
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Measurement of Pressure Differences
mambb
mmba
gRZgPP
RZgPP
)(
)(
3
2
Apply the basic equation of static fluids to both legs of manometer, realizing that P2=P3.
)( bamba RgPP
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ExampleA 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 ?
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10 Minute ProblemThe 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 ?
PG
Gas
20 in 24 in
3 in
16 in
3 in
12 in
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Extra Practice Problems
Problem Set Handout: I-52 – I-63