CHAPTER 1INTRODUCTION TO ENGINEERING CALCULATIONS

ERT 214 Material and Energy Balance / Imbangan Bahan dan Tenaga

Sem 1, 2015/2016

After completing this chapter, you should be able to do the following: 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

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

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

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 1 g = 0.036 g

  1000 mgConversion factor

Example

Convert an acceleration of 1 cm/s2 to its equivalent in km/yr2

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:

1hr=60min1min=60sSo, 1hr=3600s

1 d=24hr

Dimensional Equation

1. Write the given quantity and units on the left

2. Write the units of conversion factors that cancel the old unit and replace them with the desired unit

3. Fill the value of the conversion factors

4. Carry out the arithmetic value

Eg: Convert 1 cm/s2 to km/yr2

1 cm s2 h2 day2 m km

s2 h2 day2 yr2 cm m

1 cm 36002 s2 242 h2 3652 day2 1 m 1 km

s2 12 h2 12 day2 12 yr2 100 cm 1000 m

(3600 x 24 x 365) 2 km= 9.95 x 109 km/ yr 2

100 x 1000 yr2

TEST YOURSELF

1) What is a conversion factor?2) What is the conversion factor for s/min

(s=second)?3) What is the conversion factor for min2/s2?4) 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 minWhat is the conversion factor for min2/s2?(1min)2/(60s)2

What is the conversion factor for m3/cm3?1m3/106cm3

Systems of Units

Components of a system of units: Base units - units for the dimensions of mass,

length, time, temperature, electrical current, and light intensity.

Multiple units- multiple or fractions of base unit 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)

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.

3 Systems of Unit:

Base Units

Base Units

Quantity SISymb

olAmerican

Symbol

CGSSymb

ol

Length meter m foot ftcentime

tercm

Masskilogra

mkg

pound mass

lbm gram g

Molesgram-mole

molepound mole

lbmole

gram-mole

mole

Time second s second s second s

Temperature

Kelvin K Rankine R Kelvin K

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

JouleCalorie

Jcal

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 unitsExample: 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

Force is proportional to product of mass and acceleration (according Newton second law of motion)

kg.m/s2 (SI unit), g.cm/s2 (CGS) and lbm.ft/s2(American engineering)

Usually defined using derived units ;

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

The force in newtons required to accelerate a mass of 4.00 kg at a rate of 9m/s2 is

= 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

= 1.12lbf

4 kg 9m 1N

s2 1kg.m/s2

4 lbm 9ft 1lbf

s2 32.147lbm.ft/s2

Example

gc is used to denote the conversion factor from a natural force unit to a derived force unit.

gc = 1 kg.m/s2 = 32.174 lbm.ft/s2

1N 1 lbf

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 . 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

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.

Two-Point Linear Interpolation

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.

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:

Generate a Straight Line

THANK YOU