Transcript CHAPTER 1 Introduction to engineering calculations

CHAPTER 1
INTRODUCTION TO
ENGINEERING CALCULATIONS
Sem 1, 2015/2016
ERT 214 Material and Energy Balance / Imbangan
Bahan dan Tenaga
After completing this chapter, you should be able to do the following:
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Convert a quantity expressed in one set of units into its equivalent in any other
dimensionally consistent units using conversion factor tables.
Identify the units commonly used to express both mass and weight in SI, CGS, and
American Engineering units.
Identify the number of significant figures in a given value expressed in either decimal or
scientific notation and state the precision with which the value is known based on its
significant figures.
Explain the concept of dimensional homogeneity of equations.
Given tabulated data for two variables (x and y), use linear interpolation between two data
points to estimate the value of one variable for a given value of the other.
Given a two-parameter expression relating two variables [such as y = a sin(2x) + b or P
=1/(aQ3 + b) and two adjustable parameters (a and b), state what you would plot versus
what to generate a straight line. Given data for x and y, generate the plot and estimate the
parameters a and b.
Units and Dimensions
Objectives:
 Convert one set of units in a function or equation into another
equivalent set for mass, length, area, volume, time, energy and
force
 Specify the basic and derived units in the SI and American
engineering system for mass, length, volume, density, time, and
their equivalence.
 Explain the difference between weight and mass
 Apply the concepts of dimensional consistency to determine
the units of any term in a function
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Dimensions are:
 properties that can be measured such as length, time, mass,
temperature,
 properties that can be calculated by multiplying or dividing
other dimensions, such as velocity (length/time), volume,
density
Units are used for expressing the dimensions such as feet or
meter for length, hours/seconds for time.
Every valid equation must be dimensionally homogeneous:
that is, all additive terms on both sides of the equation must
have the same unit
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The numerical values of two quantities may be added or
subtracted only if the units are the same.
On the other hand, numerical values and their corresponding
units may always be combined by multiplication or division.
Conversion of Units
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A measured quantity can be expressed in terms of any
units having the appropriate dimension
To convert a quantity expressed in terms of one unit to
equivalent in terms of another unit, multiply the given
quantity by the conversion factor
Conversion factor – a ratio of equivalent values of a
quantity expressed in different units (new unit/old unit)
Let’s say to convert 36 mg to gram
36 mg
1g
1000 mg
=
0.036 g
Conversion
factor
Example
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Convert an acceleration of 1 cm/s2 to its equivalent in
km/yr2
1hr=60min
1min=60s
So, 1hr=3600s
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1 d=24hr
A principle illustrated in this example is that raising a
quantity (in particular, a conversion factor) to a power
raises its units to the same power. The conversion
factor for h2/day2is therefore the square of the factor
for h/day:
Dimensional Equation
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Eg: Convert 1 cm/s2 to km/yr2
1.
2.
3.
4.
Write the given quantity and units on the left
Write the units of conversion factors that cancel the old unit
and replace them with the desired unit
1 cm
s2
h2
day2
m
km
s2
h2
day2
yr2
cm
m
Fill the value of the conversion factors
1 cm
36002 s2
242 h2
3652 day2
1m
1 km
s2
1 2 h2
12 day2
12 yr2
100 cm
1000 m
Carry out the arithmetic value
(3600 x 24 x 365)
100 x 1000
2
km
yr2
=
9.95 x 109 km/ yr
2
TEST YOURSELF
1)
2)
3)
4)
What is a conversion factor?
What is the conversion factor for s/min
(s=second)?
What is the conversion factor for min2/s2?
What is the conversion factor for m3/cm3?
TEST YOURSELF
(Ans)
What is a conversion factor?
a ratio of equivalent values of a quantity expressed in different units (new
unit/old unit)
What is the conversion factor for s/min (s=second)?
60s/1 min
What is the conversion factor for min2/s2?
(1min)2/(60s)2
What is the conversion factor for m3/cm3?
1m3/106cm3
Systems of Units
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Components of a system of units:
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Base units - units for the dimensions of mass, length,
time, temperature, electrical current, and light intensity.
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Multiple units- multiple or fractions of base unit
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E.g.: for time can be hours, millisecond, year, etc.
Derived units - units that are obtained in one or two
ways;
a)
By multiplying and dividing base units; also referred
to as compound units

Example: ft/min (velocity), cm2(area), kg.m/s2 (force)
b)
As defined equivalent of compound unit (Newton = 1
kg.m/s2)
3 Systems of Unit:
a) SI system
System International – eg: meter (m) for length, kilogram (kg) for
mass and second (s) for time.
b) American engineering system
Are the foot (ft) for length, the pound mass (lbm) for mass and the
seconds (s) for time.
c) CGS system
Almost identical to SI. Principles different being that grams (g) and
centimeters (cm) are used instead of kg and m as the base units of
mass and length.
Base Units
Base Units
Quantity
SI
Symbol
American
Symbol
CGS
Symbol
Length
meter
m
foot
ft
centimeter
cm
Mass
kilogram
kg
pound mass
lbm
gram
g
Moles
grammole
mole
Time
second
s
second
s
second
s
Temperature
Kelvin
K
Rankine
R
Kelvin
K
pound mole lbmole gram-mole
mole
Base units - units for the dimensions of mass, length, time, temperature, electrical current, and light
intensity.
Multiple Units Preferences
Multiple Unit Preferences
tera (T) = 10 12
centi (c) = 10 -2
giga (G) = 10 9
milli (m) = 10 -3
mega (M) = 10 6
micro (μ) = 10 -6
kilo (k) = 10 3
nano (n) = 10 -9
Multiple units- multiple or fractions of base unit
E.g.: for time can be hours, millisecond, year, etc
Derivatives SI Units
Derived SI Units
Quantity
Unit
Symbol
Equivalent to the Base Unit
Volume
(L x W x H)
Liter
L
0.001m3 = 1000 cm3
Force
(F=ma)
Newton (SI)
Dyne (CGS)
N
1 kg.m/s2
1 g.cm/s2
Pressure
(force/area)
Pascal
Pa
1 N/m2
Energy/ Work
(E=1/2*MV2) depends on mass n speed
Joule
Calorie
J
cal
1 N.m = 1 kg.m2/s2
4.184 J =1 cal
Power
(force x distance)/time
Watt
W
1 J/s = 1 kg.m2/s3
Derived units - units that are obtained in one or two ways;
a) By multiplying and dividing base units; also referred to as compound units
Example: ft/min (velocity), cm2(area), kg.m/s2 (force)
b) As defined equivalent of compound unit (Newton = 1 kg.m/s2)
Example
Conversion Between Systems of Units
 Convert 23 Ibm. ft/min2 to its equivalent in kg·cm/s2.
Ans:
 As before, begin by writing the dimensional equation, fill
in the units of conversion factors (new/old) and then the
numerical values of these factors, and then do the
arithmetic.
TEST YOURSELF
1. What are the factors (numerical values and units) needed to convert?
(a) meters to millimeters?
(b) nanoseconds to seconds?
(c) square centimeters to square meters?
(d) cubic feet to cubic meters (use the conversion factor table on the inside
front cover)?
(e) horsepower to British thermal units per second?
2. What is the derived SI unit for velocity? The velocity unit in the CGS system?
In the American engineering system?
TEST YOURSELF
(Ans)
1. What are the factors (numerical values and units) needed to convert?
(a) meters to millimeters? 1000mm/1m
(b) nanoseconds to seconds? 10-9s/1ns
(c) square centimeters to square meters? 1m2/104cm2
(d) cubic feet to cubic meters (use the conversion factor table on the
inside front cover)? 1m3/35.3145ft3
(e) horsepower to British thermal units per second? (9.486x10-4
Btu/s)/(1.341x10-3hp)
2. What is the derived SI unit for velocity? The velocity unit in the CGS
system? In the American engineering system? m/s, cm/s , ft/s
Force and Weight
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Force is proportional to product of mass and acceleration (according
Newton second law of motion)
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kg.m/s2 (SI unit), g.cm/s2 (CGS) and lbm.ft/s2(American engineering)
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Usually defined using derived units ;
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1 Newton (N)
=
1 kg.m/s2
1 dyne
=
1 g.cm/s2
1 Ibf
=
32.174 Ibm.ft/s2
Weight of an object is force exerted on the object by gravitational
attraction of the earth i.e. force of gravity, g.
 Value of gravitational acceleration:
g
= 9.8066 m/s2
= 980.66 cm/s2
= 32.174 ft/s2
Example
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The force in newtons required to accelerate a mass of
4.00 kg at a rate of 9m/s2 is
4 kg
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9m
1N
s2
1kg.m/s2
= 36.0 N
The force in lbf required to accelerate a mass of 4.00
Ibm at a rate of 9.00 ft/s2 is
4 lbm
9ft
1lbf
s2
32.147lbm.ft/s2
= 1.12lbf
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gc is used to denote the conversion factor from a
natural force unit to a derived force unit.
gc
=
1 kg.m/s2
1N
= 32.174 lbm.ft/s2
1 lbf
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The weight of an object is the force exerted on the object by gravitational
attraction.
Suppose that an object of mass m is subjected to a gravitational force W
(W is by definition the weight of the object) and that if this object were falling
freely its acceleration would be g.
The weight, mass, and free-fall acceleration of the object are related by
W=mg
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. The value of g at sea level and 45'" latitude is given below in each system of
units:
Sample Mean
Variations in sampling and chemical analysis procedures invariably introduce scatter
in measured values (X).
We might ask two questions about the system at this point.
1. What is the true value of X?
2. How can we estimate of the true value of X?
So, we collect N (numb of measured values of X (Xl, X2, . .. , XN) and then calculate
Two-Point Linear Interpolation
Given tabulated data for two variables (x and y), use linear interpolation between two
data points to estimate the value of one variable for a given value of the other.
The equation of the line through (Xl, y1) and (X2, Y2) on a plot of y versus x is
You may use this equation to estimate y for an x between Xl and X2;
You may also use it to estimate y for an x outside of this range (i.e., to extrapolate the
data), but with a much greater risk of inaccuracy.
Generate a Straight Line
Given a two-parameter expression relating two variables [such as y = a sin(2x) + b or P
=1/(aQ3 + b) and two adjustable parameters (a and b), state what you would plot versus
what to generate a straight line. Given data for x and y, generate the plot and estimate the
parameters a and b. Example:
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