Post on 29-Jan-2016
Chapter 4 1CHEM ENG 1007
Reading Materials: Chapter 4
LECTURES 4-5
Chapter 4 2CHEM ENG 1007
Class Activity
1. How tall are you?
2. How long does it take you to get to Adelaide Uni every morning?
3. How many days are there in one year?
4. How heavy are you?
5. What is the distance from the earth to the moon?
6. What is the average weight of the new-born baby?
Chapter 4 3CHEM ENG 1007
Objective
At the end of these lectures you should be able to:
convert between sets of units by using dimensional equation
add, subtract, multiply and divide units use SI, American Engineering and cgs
units understand dimensional homogeneity
Chapter 4 4CHEM ENG 1007
Quick Quiz 1: True or FalseA quantity is meaningless without units!
§4.1 Units & Dimensions
"the distance from my house to school is six"
unless we follow that statement with "miles" or "kilometres", or whichever unit, the
above statement is meaningless.
Chapter 4 5CHEM ENG 1007
Quick Quiz 2: True or FalseAll additive/subtractive terms must have the same dimension!
§4.1 Units & Dimensions
Chapter 4 6CHEM ENG 1007
Every Monday you buy three litres of milk and five packages of Tim Tam. Since you have to carry your purchases home, you’d like to know how many kilograms of food you’ve bought.Can we just add together litres of milk plus packages of tim tam to get kilograms of food?
(3 L milk) + (5 packages of tim tam) =
? kg of food
Illustration 1
Chapter 4 7CHEM ENG 1007
a) 6.201 cm + 7.4 cm + 0.68 cm + 12.0 cm = ?b) 1.6 km + 1.62 m + 1200 cm = ?c) 8.264 g - 7.8 g = ?d) 10.4168 m - 6.0 m = ?e) 12.00 m + 15.001 kg = ?f) 131 cm x 2.3 cm = ?g) 5.7621 m x 6.201 m = ?h) 20.2 cm divided by 7.41 s = ?i) 40.002 g divided by 13.000005 ml = ?
Illustration 2
Assignment 1, Question 7
Chapter 4 8CHEM ENG 1007
Case Study 1: Feet or Meters?
SEOUL, South Korea – A mix up in the cockpit over whether altitude guidance was measured in feet or meters led to the crash of a Korean Air Lines McDonnell Douglas MD-11 freighter soon after takeoff in Shanghai in April 1999.
The crash killed all 3 crew-members. Five people on the ground were killed and 40 more were injured when the plane went down in light rain onto a construction site near Shanghai’s Hongqiao Airport.
Chapter 4 9CHEM ENG 1007
Wall Street Journal, June 6, 2001
A Chinese air-traffic controller directed the pilots to an altitude of 1,500 meters (4,950 feet). The plane was climbing rapidly to that level when the co-pilot told the pilot he thought the instructed height was 1,500 feet (455 meters). The international aviation industry commonly measures altitude in feet, and the confusion led the pilot to conclude the jet was almost 1,000 meters too high, so he quickly moved the controls to lower the plane. As the plane descended, the pilot realized the error but couldn’t correct the mistake in time.
South Korea’s Ministry of Construction and Transportation said Korean Air Lines would lose the right to serve the Seoul-Shanghai cargo route for at least two years because of errors by the pilots. Korean Air Lines said it would appeal the decision…..
Case Study 1: Feet or Meters?
Chapter 4 10CHEM ENG 1007
Mars Probe Lost Due to Simple Math Error
NASA lost its $125-million Mars Climate Orbiter because spacecraft engineers failed to convert from English to metric measurements when exchanging vital data before the craft was launched.
pound-seconds versus newton-seconds
Want to know more…..
http://articles.latimes.com/1999/oct/01/news/mn-17288
Case Study 2
Chapter 4 11CHEM ENG 1007
Dimensions versus Units
Dimensions are properties that can be measured.
Four basic dimensions: length [L], mass [M], time
[T], and temperature []
Units are specific magnitudes of dimensions.
Different units can measure the same dimension
Example: Dimension: length
Units: foot, metre, yard, mile, etc.
Such units are related by a Conversion factor
Chapter 4 12CHEM ENG 1007
Dimension versus units
Dimension: mass
Units: kilogram (kg) and pound (lbm)
0
1 kg
1 lbm
0.454 kg 2 kg
2 lbm
Conversion factor: 1 lbm = 0.453593 kg
1 kg = 2.2046 lbm
Illustration 3
Chapter 4 13CHEM ENG 1007
Systems of Units
Three Base Unit groupings are:
1. Metric system (also known as SI system)
2. American / British engineering system.
3. CGS (Centimetres-Grams-Seconds) system.
Should be
Kelvin
Chapter 4 14CHEM ENG 1007
Concept Inventory
1. Which of the following set of units is in CGS system
a) kg, m, s, N
b) lbm, ft, s, lbf
c) g, m, s, N
d) g, cm, s, dyne
e) kg, cm, s, dyne
Chapter 4 15CHEM ENG 1007
§4.1.1 Conversion Factors
A conversion factor is an equation that equates two quantities expressed in different units where, in fact, the two quantities are exactly equivalent.
Examples:
12 in = 1 ft 1000 g = 1 kg 60s = 1 min
Chapter 4 16CHEM ENG 1007
Conversion Factors Table
Or turn to page xi of the text-book
Chapter 4 17CHEM ENG 1007
§4.1.1 Conversion Factors
Conversion factors dimensionless quantities.
Enable conversion between different units systems.
Conversion factors do not change magnitude, just value of units.
Chapter 4 18CHEM ENG 1007
Is Conversion factor of metres to feet really dimensionless?
1m 3.281ft
Rearrange?: Dimensionless?:
ft
1 3.281m
[L]ft
3.281m [L]
Dimensionless
Illustration 4
Chapter 4 19CHEM ENG 1007
PrefixesPrefixes are used in SI to indicate powers of ten. Multiples or fractions of basic units defined for convenience.
Factor Prefix Symbol Factor Prefix Symbol
109 giga G 10-1 deci d
106 mega M 10-2 centi c
103 kilo k 10-3 milli m
102 hecto h 10-6 micro
101 deka da 10-9 nano n
Commonly used for Joule, Pascal, Metre, Byte
Chapter 4 20CHEM ENG 1007
Quick Quiz
How many Megabytes in 1Gigabytes?
From the prefixes table, we get the following conversion factors
109 Bytes = 1 GB
106 Bytes = 1 MG
By dimensional equation
1 GB910 Bytes
1 GB 6
1 MB
10 Bytes 310 MB
Chapter 4 21CHEM ENG 1007
Quick Quiz
How many micrograms in 5 kilograms?
Again from the prefixes table, we get the following conversion factors
103 g = 1 kg
10-6 g = 1 g
5 kg 310 g
1 kg
6
1 g
10 g 95x10 g
Chapter 4 22CHEM ENG 1007
Quick Quiz
How many millimetres in 1 nanometres?
Again from the prefixes table, we get the following conversion factors
10-3 m = 1 mm
10-9 m = 1 nm
1 nm 9 10 m
1 nm
3
1mm
10 m 610 mm
Chapter 4 23CHEM ENG 1007
Write conversion factors that relate to each of the following pairs of units:
1. Litres and mL
2. Hours and seconds
3. mg and kg
4. Nanometers and kilometers
Quick Quiz
1 L = 1000 mL
1 h = 3600 s
106 mg = 1 kg
1012 nm = 1 km
Chapter 4 24CHEM ENG 1007
Unit Conversion
Conversion of units from one system to another is necessary in process calculations and analysis, when
Data is from different sources, or
Variables are measured from instruments of different standards
(starting quantity) x (conversion factor) = equivalent quantity
Chapter 4 25CHEM ENG 1007
Illustration 5
Convert 28 inches to its equivalent number of feet. 12 in = 1 ft
28 in 1
12
ft
in
2.333
12 28
1
ft = 2.3 ft (s.f)
inin
ft
When solving a problem, the idea is to set up an equation so that all unwanted units cancel, leaving only the desired units.
Convert mass of 60 lbm into kg?
1 lbm = 0.454 kg
60 mlb 0.454
m
kg
lb
27 30 kg (1 s.f )kg
Chapter 4 26CHEM ENG 1007
Dimensional Equation A convenient method for unit conversion.How to set up a dimensional equation?
Write the given quantity and its units on the left, write the units of conversion factors that
cancel the old units and replace them with the described ones, fill in the values of the conversion factors, and carry out the
indicated arithmetic to find the desired value.
28in ft
in
Chapter 4 27CHEM ENG 1007
Convert a mass of 150 lbm into equivalent in kg.
Solution:
If 1 lbm is equivalent to 0.454 kg then
or by dimensional equation
m m
m
0.454kg150 lb 150 lb 68.1kg 68 kg (2 s.f )
1 lb
mm
150 lb150 lb
m
0.454 kg
1 lb68 kg
Illustration 6
Chapter 4 28CHEM ENG 1007
Using dimensional equation, convert a weight 90 N into lbf ?
1 N = 0.2281 lbf
90 N90 N f0.2281lb
N f f20.3 lb 20 lb
8 m 3.2808 ft 60 sm ft ft
8 1575 2000 (1 s.f )s m 1mins min min
Illustration 7
Using dimensional equation, convert 8 m/s into ft/min?
1 m = 3.2808 ft; 1 min = 60 s
Chapter 4 29CHEM ENG 1007
Convert a 40 mile/gal into km/L
Convert a force of 40 lbf into equivalent in N.
mile 40 mile
40 gal gal
1.6 km
1 mile
1 gal km14.2
4.5 L L
ff
40 lb40 lb
f
1 N
0.225 lb178 N
Quick Quiz
Chapter 4 30CHEM ENG 1007
Derived as combination of the base units: by multiplying or dividing of the base units
velocity (m/s), area (m2), flow rate (ft3s-1)
§4.1.5 Combined Units
Chapter 4 31CHEM ENG 1007
§4.1.5 Combined Units
To produce conversion factors for combined units,
conversion factors of base units can be raised to
any power.
Examples:
2 4
3
100 10area :
1 1
12 1728volume :
1 1
22
2
33
3
cm cm L
m m
in inL
ft ft
Chapter 4 32CHEM ENG 1007
Convert 25 lbm.ft/min2 to its equivalent in kg.cm/s2?
2 m2
m 22m
m
25 lb .ft 0.4536 kg 30.48 cm 1 min25 lb .ft / min
min 1 lb 1 ft 60 s
25 lb
. ft 20.4536 kg 30.48 cm 1 min
2min m1 lb 1 ft 2 2
2
60 s
= 0.096 kg cm/s
Illustration 8
Chapter 4 33CHEM ENG 1007
Convert 45 m3/d2 to its equivalent in mm3/s2?
3 2 23
3 23 2 22
45 m 1000 mm 1d 1h45 m / d
d 1 m 24h 3600 s
Quick Quiz
3 33 2 2
2 2 2 22 3
45 m 1000 mm 1 d 1h
d 1 m 24 h 3600 s
345 m 3 3 21000 mm 1 d 21h
2d 31 m 2 224 h
2 2
3 2
3600 s
6.0 mm /s
Chapter 4 34CHEM ENG 1007
Derived equivalents of combined units:
1 N = 1 kgm s-2
1 dyne = 1 gcm s-2
1 lbf = 32.174 lbmft s-2
1 Pa = 1 Nm s-2
1 gal = 0.16 ft3 (imperial)
1 gal = 0.133 ft3 (US)
§4.1.5 Combined Units
Coherent derived units
Non-coherent derived units
Chapter 4 35CHEM ENG 1007
Coherent versus Non-coherent derived Units
Coherent Derived Units:
Value of defined unit = Combined value of Basic Units
e.g. 1 J (joule) = 1 (kg.m2)/s2
Non-coherentValue of defined unit Value of base units
e.g. 1 Btu 1 (kg.m2)/s3 Conversion factor required
e.g. 9.486x10-4 Btu = 1 (kg.m2)/s3 Many American/British defined units are non-coherent (e.g. Btu, lbf)
Chapter 4 36CHEM ENG 1007
Summary for Units Conversion
To convert a quantity expressed in terms of one unit to its equivalent in terms of another unit, multiply the given quantity by the conversion factor (new unit/old unit).
1 g45 mg x 0.045 g
1000 mg
45 mg 1g
1000 mg0.045 g
Alternative way:
Chapter 4 37CHEM ENG 1007
Rules of dimensional consistency
§4.2.5 Dimensional Consistency
1. Terms that are added together (or subtracted) must have the same units. For example, in the equation Q = ab + c2, the units of ab must be the same as those of c2.
2. Exponents must be unitless (dimensionless). Thus, if an exponent consists of several terms, the units of all those terms must cancel. For example, in the equation y = xab/c, the units in the term ab/c must all cancel out to leave no units.
Chapter 4 38CHEM ENG 1007
Arrhenius equation
Illustration 9
Parameters Symbols Units
Gas constant R J / mol.K
Temperature T K
Activation energy Ea ?
aEk A exp
RT
a aE E
RT J
mol.K
K
a
1
JE
mol
Chapter 4 39CHEM ENG 1007
Ideal Gas Equation
Illustration 10
Parameters Symbols Dimensions Units
Pressure P [M] [L]-1 [T] -2 Pa
Volume V [L]3 m3
# of moles n [N] mol
Gas constant R [M] [L] 2 [T] -2 [N]-1 []-1 Pa.m3/mol.K
Temperature T [] K
PV nRT
In terms of these parameters, show that ideal gas equation obeys the laws of dimensional consistency
Chapter 4 40CHEM ENG 1007
Substitute units into Equation?
3LHS P V Pa . m
RHS n R T mol
3Pa.m
mol
.K K
3Pa . m
i.e., LHS = RHS
Illustration 10
Chapter 4 41CHEM ENG 1007
Substitute base dimensions into Equation?
23
2 2
[M] [M] [L]LHS . [L]
[L] [T] [T]
RHS [N]
2[M] [L]
. [N] [ ] 2
. [ ][T]
2
2
[M] [L]
[T]
LHS RHS
Illustration 10
Chapter 4 42CHEM ENG 1007
Consider this equation dv
dy
2shear stress N/m
v velocity m / s
y width m
What are SI units of ?
Illustration 11
Chapter 4 43CHEM ENG 1007
Compare dimensional homogeneity!
Rearrange yields units for .
2
N mLHS .
m
/ s
m
RHS
2
N . sPa.s
m
Illustration 11
dv
dy
Chapter 4 44CHEM ENG 1007
§4.1.6 Force
The force “F” to accelerate a mass “m” at an acceleration rate “a” is defined by Newton’s second law as
F = ma [M.L.T-2]
One form of acceleration is that associated with earth’s gravity. The rate of acceleration is described by the gravitational acceleration, g, and the force that an object exerts on the earth’s surface (its “weight”), is
W = mg
Chapter 4 45CHEM ENG 1007
§4.1.6 Force
Factors for Unit conversions of Force
1 N = 1 kg.m/s2 = 105 dynes = 105 g.cm/s2 = 0.22481 lbf
Chapter 4 46CHEM ENG 1007
Example 4.2
mm 2 2
m ff2
m2
lb ft32.174ftW mg 1 lb 32.174
s s
lb ft 1 lbW 32.174 1 lb
s lb ft32.174
s
An object has a mass equal to 1 lbm. What is its weight in pounds-force (lbf)
Solution:
From Table 4.3, we note that g = 32.174 ft/s2
Chapter 4 47CHEM ENG 1007
Example 4.3
2 2
2
2
a)
9.8066 m kg mW mg 8.41 kg 82.5
s s
kg m 1 NW 82.5 82.5 N
kg ms 1 s
An object has a mass equal to 8.41 kg. What is its weight
a) in Newtons (N)
b) pounds-force (lbf)
Solution:
ff
b)
0.22481 lbW 82.5 N 18.5 lb
1 N
Chapter 4 48CHEM ENG 1007
Quick Quiz
From Reading Question 4:
The weight of an astronaut is measured on a distant planet and found to be one-fifth of his weight on the earth’s surface.
Is his mass different on that distant planet than on earth?
What does the weight difference imply about the acceleration of gravity on the distant planet?
Chapter 4 49CHEM ENG 1007
Summary2 case-studies
Identified the difference in SI, American Engineering, and CGS units system
Understand conversion factors dimensional equations dimensionless dimensional consistency coherent and non-coherent derived units Force/Weight
Able to convert between sets of units