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Introduction to Chemical Engineering Calculations
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños
Lecture 1.
Units and Dimensions
Mathematics and Engineering
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE
2
In mathematics,
If x = 500 and y = 100, then (x + y) = 600
In engineering,
If x = 500m and y = 100m, then (x + y) = 600m
But,
If x = 500m and y = 100kg, then (x + y) = 600???
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE
3
Why Do We Need Units?
Units are important for effective communication and standardization of measurements
Image Source: http://lamar.colostate.edu/~hillger/feet-to-meters_1.jpg
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE
4
The “Gimli Glider” Incident (23 July 1983)
Image Source: http://upload.wikimedia.org
Pounds vs Kilograms
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE
5
The Mars Climate Orbiter Incident (23 September 1999)
Newton vs Pound Force
Image Source: http://www.jpl.nasa.gov/... /climate-orbiter-browse.jpg
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE
6
Dimension Symbol
Mass m
Length L
Time θ
Temperature T
Mole n
Luminosity c
Electric Current I
The 7 Fundamental (Base) Dimensions
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE
7
Dimension SI Unit Am. Eng. Unit
Masskilogram
(kg)poundmass
(lbm)
Lengthmeter(m)
foot(ft)
Timesecond
(s)second
(s)
TemperatureKelvin
(K)Rankine
(0R)
Molegram mole
(gmol)pound mole
(lbmol)
SI and American Engineering System Units for Fundamental Dimensions
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE
8
Dimension Symbol
Area L2
Volume L3
Velocity L/θ
Acceleration L/θ2
Force m· (L/θ2)
Pressure m· (L/θ2)/L2 = m/θ2· L
Energy m· (L/θ2)· L = m· (L2/θ2)
Secondary (Derived) Dimensions
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE
9
Dimension SI Unit Am. Eng. Unit
Volume m3 ft3
Acceleration m/s ft/s
Force kg · m/s2 lbm · ft/s2
Pressure kg /(m · s2) lbm /(ft · s2)
Energy kg· (m2/s2) lbm· (ft2/s2)
SI and American Engineering System Units for Secondary Dimensions
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE10
Dimension SI Unit Am. Eng. Unit
Force1kg · m/s2
= 1 N32.174 lbm · ft/s2
= 1 lbf
Pressure1 kg /(m · s2)= 1 N/m2 = 1
Pa
32.174 lbm /(ft · s2)= 1 lbf/ft2
= (1/144) lbf/in2 (psi)
Energy1 kg· (m2/s2)= 1 N· m = 1 J
32.174 lbm· (ft2/s2)= 1 ft · lbf
Defined Equivalent Units
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE11
Conversion of Units: Single Measurements
The equivalence between two units of the same measurement may be defined in terms of a ratio (conversion factor):
New UnitOld Unit = New Unit Old Unit
1 Old Unit 1=Old Unit New Unit New Unit
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE12
Conversion of Units: Single Measurements
2 2
2 2 2 2 2
2 2 29
2 2
2.2 lbm500 kg =1100 lbm kg
300 1 cm 300 1 cm 3= = 10 mmcm cm 10 mm mm
cm 3600s 24h 365d 1m 1km km1 = 9.95x101h 1d 1yr 100cm 1000ms yr
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE13
Conversion of Units: Equations or Formula
Consider the following equation of motion:
D (ft) = 3 t(s) – 4
Derive an equivalent equation for distance in meters and time in minutes.
Step 1. Define new variables D’(m) and t’(min).
Step 2. Define the old variables in terms of the new variable.
Units and Dimensions1
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE14
Conversion of Units: Equations or Formula
3.2808 ftD(ft) = D'(m) x or D = 3.2808D'1 m
60 st(s) = t'(min) x or t = 60t'1 min
Step 3. Substitute these equivalence relations into the original equation.
(3.2808D’) = 3 (60t’) – 4
Simplifying,
D’ (m) = 55t’(min) – 1.22
Units and Dimensions1
The numerical value of two or more quantities can be added/subtracted only if the units of the quantities are the same.
5 kilograms + 3 meters = no physical meaning
10 feet + 3 meters = has physical meaning
10 feet + 9.84 feet = 19.94 feet
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE15
Operation on Units: Addition and Subtraction
Units and Dimensions1
Multiplication and division can be done on quantities with unlike units but the units can only be cancelled or merged if they are identical.
5 kilograms x 3 meters = 15 kg-m
3 m2/60 cm = 0.05 m2/cm
3 m2/0.6 m = 5 m2/m = 5 m
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE16
Operation on Units: Multiplication and Division
Units and Dimensions1
Reynolds Number Calculation
Reynolds number is calculated as:
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE17
Dimensionless Quantities
ρDvReynolds Number = µ
Units and Dimensions
where = density of the fluid (kg/m3)D = diameter of pipe (m)v = mean velocity of fluid (m/s) = dynamic viscosity (kg/m· s)
What is the net dimension of Reynolds Number?
1
Importance of Dimensionless Quantities
Used in arguments of special functions such as exponential, logarithmic, or trigonometric functions.
e20 is possible but e(20ft) is undefined
cos(20) is possible but cos(20 ft) is undefined
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE18
Dimensionless Quantities
Units and Dimensions1
Importance of Dimensionless Quantities
Consider the Arrhenius Equation:
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE19
Dimensionless Quantities
aE-RTk = Ae
Units and Dimensions
If Ea is activation in cal/mol and T is temperature in K, what is the unit of R?
To make the argument of the exponential function dimensionless, R must have a unit of (cal/mol-K).
1
Every valid equation must be dimensionally consistent.
Each term in the equation must have the same net dimensions and units as every other term to which it is added, subtracted, or equated.
A + B = C – DE
If A has a dimension of L3, then
1. B must have a dimension of L3 since it is added to A.2. (A + B) has a net dimension of L3.3. (C – DE) must have a net dimension of L3
4. C and DE have a dimension of L3.
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE20
Dimensional Consistency
Units and Dimensions1
Units and Dimensions
Example on Dimensional Consistency
The density of a fluid is given by the empirical equation
= 70.5 exp(8.27 x 10-7 P)
where = density in (lbm/ft3) and P = pressure (lbf/in2).
a. What are the units of 70.5 and 8.27x10-7?
b. Derive a formula for (g/cm3) and P (N/m2)
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE21
Dimensional Consistency
1
Units and Dimensions
= 70.5 exp(8.27 x 10-7 P)
1. Since the exponential part is dimensionless, then 70.5 must have the same unit as which is (lbm/ft3).
2. Since the argument of the exponential function must be dimensionless, then 8.27 x 10-7 must have a unit of (in.2/lbf) which is a reciprocal to the unit of P.
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE22
Dimensional Consistency
1
Units and Dimensions
1. Define new variables ’ (g/cm3) and P’ (N/m2).
2. Express the old variables in terms of the new variables.
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE23
Dimensional Consistency
3
3 3 3
2-4
2 2 2
lbm g 1 lbm 28,317 cmρ = ρ' = 62.43ρ'453.593 gft cm 1 ft
lbf N 0.2248 lbf 1 mP = P' = 1.45x10 P'1 Nin m 39.37 in
1
3. Substitute the equivalence relations into the original equation.
Prof. Manolito E Bambase Jr. Department of Chemical Engineering. University of the Philippines Los Baños SLIDE24
Dimensional Consistency
Units and Dimensions
-7
-7 -4
ρ = 70.5 exp 8.27 x 10 P
62.43ρ' = 70.5 exp 8.27 x 10 1.45 x 10 P'
Simplifying,
-103 2
g Nρ' = 1.13 exp 1.20 x 10 P'cm m
1