CHE1003:: FLUID FLOW & FLOW MEASUREMENT

CHE1003:: FLUID FLOW & FLOW MEASUREMENT

Class Timing

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Course Objective: • Understand fundamental concepts of Fluid flow and to apply those concepts

to real engineering problems. • Understand the overview of fluid flow • Gather knowledge of Fluid statics , Kinematics and Flow measuring device

CHE1003:: FLUID FLOW & FLOW MEASUREMENT

Prerequisites: Nil

Course Details:Unit 1: Introduction to Fluid mechanics

Unit 2: Flow measurements and Fluid kinematics

Unit 3: Flow of incompressible, compressible flows and immersed bodies

Unit 4: Fluidization and its applications

Unit 5: Transportation of fluids

3

Theory Subject

(100 marks)

Surprise test

20 marks

Scaling

Class Test

Home assignments

Tutorial Quiz

Assignment

10 marks

Scaling

Experiential Learning

/Case Study

/Course Project/Skilling

20 marks Scaling

Mid Sem Examination

20 marks Scaling

End Sem examination

30 marks Scaling

Assessment Strategy

75 % attendance is mandatoryAssignment and quiz should be

submitted in assign timeOtherwise no evaluation

Student must choose one real life project related to PMS sub. Must use MATLAB .At the end of The semester they

must give presentation

CHE1003:: FLUID FLOW & FLOW MEASUREMENT

https://nptel.ac.in/courses/103/104/103104043/Course Name

Fluid MechanicsDepartment Chemical EngineeringIIT KanpurInstructor Dr. Nishith Verma

https://ocw.mit.edu/courses/mechanical-engineering/2-06-fluid-dynamics-spring-2013/index.htm

https://youtube.com

What is Fluid?

5

First understand Fluids Any Substance which can flow

Gasses and liquids

Honey Sweet PourMilk Pouring in BottleWater flowing

All Are Fluids

6

First understand Fluids Any Substance which can flow

Gasses and liquids

What is the meaning of that …?

Water molecules arecontinuously moving

Water molecules constantlychanging its relative position

All molecules are in Relative Motion

7

First understand Fluids Any Substance which can flow

Gasses and liquids What is the meaning for that …?

Water molecules arecontinuously moving

Water molecules constantlychanging its relative position

All molecules are in Relative Motion

Any Substance which can flow means All molecules should have Relative Motion, Then we can call as Fluid.

8

Gasses and liquids

Water molecules arecontinuously moving

Water molecules constantlychanging relative position

All molecules are in Relative Motion

Any Substance which can flow means All molecules should have Relative Motion, Then we can call as Fluid.

Can we say Sugar and Salt as Fluids?

No, Molecules position does not change continuously

Sugar and Salt are not as Fluids

9

Comparison of Solid, Liquid and Gas ?

10

Mechanics The branch of science in which we study about

force and its effects

Magnitude of force Direction of force

Force produce Acceleration

In Mechanics we study about force and its effects

Football kick

11

Fluid Mechanics The branch of science in which we study about

behavior of fluids under action of force

Fluids Liquids and Gasses Mechanics Motion and its cause

Three Parts

Mechanics applied to rigid body “Engineering Mechanics”

applied to flexible body “Strength of Materials”

applied to Fluid “Fluid Mechanics”

Fluid Statics Fluid Kinematics Fluid Dynamics

Study of fluid at restStudy of fluid in motion

Neglecting Pressure force

Study of fluid in motionConsidering

Pressure force

Scope of Fluid Mechanics:

Fluid Mechanics is the study of the behavior of fluids at rest and in motion .

Fluid Mechanics: Why Fluid Mechanics is A Core Course?

❑ Please try to think or name any

➢ An Industry ➢ A piece of machinery

➢ Any engineering system

➢ Human Body

Fluid Mechanics is Everywhere!

12

Application of Fluid Mechanics

Industries where Concepts of Fluid Mechanics Play a Vital Role

Auto-mobile Engineering Power Generation Industry Space or Rocket Industry

Aerospace EngineeringChemical Engineering Environmental Control

And Many More 13

Fluid Mechanics The branch of science in which we study about

behavior of fluids under action of force

Fluid is a substance that deforms continuously under the application (or the effect) of a shear stress(Shear force/application area).

Mechanics applied to Fluid “Fluid Mechanics”

Shear force tangential force

Force Perpendicular to Surface

Normal force

Stress = force/application area.

14

Fluid Mechanics The branch of science in which we study about

behavior of fluids under action of force

Fluid is a substance that deforms continuously under the application (or the effect) of a shear stress(Shear force/application area).

Mechanics applied to Fluid “Fluid Mechanics”

When a shear stress

is applied

Solid body Solids deform or bend, then stop!

applied to Fluid Fluids continuously deform i.e., they Flow

15

Fluid Properties

A fluid is said to be Newtonian fluid if shear stress is directly proportional to rate of shear strain or velocity gradient.

Fluid Properties

A fluid is said to be Newtonian fluid if shear stress is directly proportional to rate of shear strain or velocity gradient.

17

A microscopic view

Gas

Fluidcompressible

Liquid

FluidIncompressible

Solid

rigid body

What new physics is involved?

• Fluids can flow from place-to-place

• Their density can change if they are compressible (for example, gasses)

• Fluids are pushed around by pressure forces

• An object immersed in a fluid experiences buoyancy

Density

𝑑𝑒𝑛𝑠𝑖𝑡𝑦 =𝑚𝑎𝑠𝑠

𝑣𝑜𝑙𝑢𝑚𝑒𝜌 =

𝑚

𝑉𝑈𝑛𝑖𝑡𝑠 𝑎𝑟𝑒

𝑘𝑔

𝑚3

• The density of a fluid is the concentration of mass

• Mass = 100 g = 0.1 kg

• Volume = 100 cm3 = 10-4 m3

• Density = 1 g/cm3 = 1000 kg m3

0%0%0%0%0%

1 2 3 4 5

The shown cubic vessels containthe stated matter. Which fluid hasthe highest density ?

1 𝑘𝑔 of water at 73°𝐶

1 𝑘𝑔 of water at 273°𝐾

1 𝑘𝑔 of water at 273°𝐶

1 𝑘𝑔 of water at 373°𝐶

1 𝑚

1 𝑚

1 𝑚

3𝑚

𝟏.

𝟑.

𝟐.

𝟒.

Pressure

• Pressure is the concentration of a force – the force exerted per unit area

Greater pressure!(same force, less area)

Exerts a pressure on thesides and through the fluid

Pressure

𝑝 =𝐹

𝐴

• Units of pressure are N/m2 or Pascals (Pa) – 1 N/m2 = 1 Pa

• Atmospheric pressure = 1 atm = 101.3 kPa = 1 x 105 N/m2

0%0%0%0%

1. 2. 3. 4.

𝑣𝑎𝑐𝑢𝑢𝑚

What is responsible forthe force which holdsurban climber B in placewhen using suction cups.

AB

1. The force of friction

2. Vacuum pressure exerts a pullingforce

3. Atmospheric pressure exerts apushing force

4. The normal force of the glass.

Hydrostatic Equilibrium

• Pressure differences drive fluid flow

• If a fluid is in equilibrium, pressure forces must balance

• Pascal’s law: pressure change is transmitted through a fluid

Hydrostatic Equilibrium with Gravity

Pressure in a fluid is equal to the weight of the fluid per unit area above it:

𝑃 = 𝑃0 + 𝜌𝑔ℎ

𝑃 + 𝑑𝑃 𝐴 − 𝑝𝐴 = 𝑚𝑔

𝑑𝑃 𝐴 = 𝜌 𝐴 𝑑ℎ 𝑔

𝑑𝑃

𝑑ℎ= 𝜌𝑔

Derivation:

𝑃 = 𝑃0 + 𝜌𝑔ℎ

0%0%0%0%0%

1. 2. 3. 4. 5.

Consider the three open containers filled with water.How do the pressures at the bottoms compare ?

1. 𝑷𝑨 = 𝑷𝑩 = 𝑷𝑪

2. 𝑷𝑨 < 𝑷𝑩 = 𝑷𝑪

3. 𝑷𝑨 < 𝑷𝑩 < 𝑷𝑪

4. 𝑷𝑩 < 𝑷𝑨 < 𝑷𝑪

5. Not enough information

A. B. C.

0%0%0%0%0%

1 2 3 4 5

The three open containers are now filled with oil,water and honey respectively. How do the pressuresat the bottoms compare ?

1. 𝑷𝑨 = 𝑷𝑩 = 𝑷𝑪

2. 𝑷𝑨 < 𝑷𝑩 = 𝑷𝑪

3. 𝑷𝑨 < 𝑷𝑩 < 𝑷𝑪

4. 𝑷𝑩 < 𝑷𝑨 < 𝑷𝑪

5. Not enough information

A. B. C.

oil water

honey

Calculating Crush Depth of a SubmarineQ. A nuclear submarine is rated to withstand a pressure differenceof 70 𝑎𝑡𝑚 before catastrophic failure. If the internal air pressureis maintained at 1 𝑎𝑡𝑚, what is the maximum permissible depth ?

𝑃 = 𝑃0+ 𝜌𝑔ℎ

𝑃 − 𝑃0 = 70 𝑎𝑡𝑚 = 7.1 × 106 𝑃𝑎 ; 𝜌 = 1 × 103 𝑘𝑔/𝑚3

ℎ =𝑃 − 𝑃0

𝜌𝑔=

7.1 × 106

1 × 103 × 9.8= 720 𝑚

Measuring Pressure

Atmospheric pressure can support a 10 meters highcolumn of water. Moving to higher density fluidsallows a table top barometer to be easily constructed.

𝑝 = 𝑝0 + 𝜌𝑔ℎ

Q. What is height of mercury (Hg) at 1 𝑎𝑡𝑚 ?

𝜌𝐻𝑔 = 13.6 𝑔/𝑐𝑚3

𝑃 = 𝑃0+ 𝜌𝑔ℎ → ℎ = 𝑃/𝜌𝑔

ℎ =1 × 105

1.36 × 104 × 9.8= 0.75 𝑚

Measuring Pressure

𝑝 = 𝑝0 + 𝜌𝑔ℎ

Q. What is height of mercury (Hg) at 1 𝑎𝑡𝑚 ?

𝜌𝐻𝑔 = 13.6 𝑔/𝑐𝑚3

Pascal’s Law

Q. A large piston supports a car.The total mass of the piston andcar is 3200 𝑘𝑔. What force mustbe applied to the smaller piston ?

Pressure at the same height is the same! (Pascal’s Law)

𝐹1

𝐴1

=𝐹2

𝐴2

𝐹1 =𝐴1

𝐴2

𝑚𝑔 =𝜋 × 0.152

𝜋 × 1.202× 3200 × 9.8 = 490 𝑁

A1

A2F2

• Pressure force is transmitted through a fluid

Gauge Pressure

Gauge Pressure is the pressure difference from atmosphere. (e.g. Tyres) 𝑃𝑎𝑏𝑠𝑜𝑙𝑢𝑡𝑒 = 𝑃𝑎𝑡𝑚𝑜𝑠𝑝ℎ𝑒𝑟𝑒 + 𝑃𝑔𝑎𝑢𝑔𝑒

Archimedes’ Principle and Buoyancy

Why do some things float and other things sink ?

Archimedes’ Principle and Buoyancy

➢ A body at rest in a fluid is acted upon by a force pushingupward called the buoyant force, which is equal to theweight of the fluid that the body displaces.

➢ If the body is completely submerged, the volume offluid displaced is equal to the volume of the body.

➢ If the body is only partially submerged, the volume ofthe fluid displaced is equal to the volume of the part ofthe body that is submerged.

Archimedes’ Principle and Buoyancy

The Buoyant Force is equal to the weight of the displaced fluid !

𝐹𝐵 = 𝑚𝑤𝑎𝑡𝑒𝑟 𝑔 = 𝜌𝑤𝑎𝑡𝑒𝑟𝑉𝑔

𝑊 = 𝑚𝑠𝑜𝑙𝑖𝑑 𝑔 = 𝜌𝑠𝑜𝑙𝑖𝑑𝑉𝑔

Objects immersed in a fluid experience a Buoyant Force!

Archimedes’ Principle and Buoyancy

The hot-air balloon floats because the weight of air displaced (= the buoyancy force) is greater than the weight of the balloon

The Buoyant Force is equal to the weight of the displaced fluid !

0%0%0%0%0%

1 2 3 4 5

Which of the three cubes of length 𝑙 shown belowhas the largest buoyant force ?

A. B. C.

water stone wood

water FB FB FB

m1g m2g m3g

1. 𝒘𝒂𝒕𝒆𝒓

2. 𝒔𝒕𝒐𝒏𝒆

3. 𝒘𝒐𝒐𝒅

4. 𝒕𝒉𝒆 𝒃𝒖𝒐𝒚𝒂𝒏𝒕 𝒇𝒐𝒓𝒄𝒆 𝒊𝒔 𝒕𝒉𝒆 𝒔𝒂𝒎𝒆

5. Not enough information

Example Archimedes’ Principle and Buoyancy

The Buoyant Force is equal to the weight of the displaced fluid.

Q. Find the apparent weight of a 60 𝑘𝑔 concrete block when you lift it under water, 𝜌𝑐𝑜𝑛𝑐𝑟𝑒𝑡𝑒 = 2200 𝑘𝑔/𝑚3

mgw =

bF

Develop

Assess

Interpret

Water provides a buoyancy force

Evaluate

Apparent weight should be lessapparentbnet wFmgF =−=

gmF waterdispb = Vgwater=

V

mcon =

con

waterapp

mgmgw

−=

con

mV

=

)1(con

waterapp mgw

−=

N321)2200

10001(8.960 =−=

Floating Objects

Q. If the density of an iceberg is 0.86that of seawater, how much of aniceberg’s volume is below the sea?

𝐵𝑢𝑜𝑦𝑎𝑛𝑐𝑦 𝑓𝑜𝑟𝑐𝑒 𝐹𝐵 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑤𝑎𝑡𝑒𝑟 𝑑𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑑

𝑉𝑠𝑢𝑏 = 𝑠𝑢𝑏𝑚𝑒𝑟𝑔𝑒𝑑 𝑣𝑜𝑙𝑢𝑚𝑒

𝐹𝐵 = 𝑚𝑤𝑎𝑡𝑒𝑟 𝑔 = 𝜌𝑤𝑎𝑡𝑒𝑟𝑉𝑠𝑢𝑏𝑔

𝐼𝑛 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚, 𝐹𝐵 = 𝑤𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑖𝑐𝑒𝑏𝑒𝑟𝑔

𝐹𝐵 = 𝑚𝑖𝑐𝑒 𝑔 = 𝜌𝑖𝑐𝑒 𝑉𝑖𝑐𝑒 𝑔

𝜌𝑤𝑎𝑡𝑒𝑟𝑉𝑠𝑢𝑏𝑔 = 𝜌𝑖𝑐𝑒𝑉𝑖𝑐𝑒𝑔 →𝑉𝑠𝑢𝑏

𝑉𝑖𝑐𝑒

=𝜌𝑤𝑎𝑡𝑒𝑟𝜌𝑖𝑐𝑒

= 0.86

Engineering Problems and Fundamental Dimensions

• when someone asks you how old you are, you reply bysaying “I am 19 years old.”

• You don’t say that you are approximately 170,000 hours oldor 612,000,000 seconds old, even though these statementsmay very well be true at that instant!

Engineering Problems and Fundamental Dimensions

• fundamental or base dimensions to correctly express whatwe know of the natural world. They are length, mass, time,temperature, electric current, amount of substance, andluminous intensity.

Systems of Units

The most common systems of units are :

➢ International System (SI) .

➢ British Gravitational (BG) .

➢ U.S. Customary units.

International System (SI) of Units

International System (SI) of Units

International System (SI) of Units

• The units for other physical quantities used inengineering can be derived from the base units.

• For example, the unit for force is the newton. It isderived from Newton’s second law of motion.

• One newton is defined as a magnitude of a force thatwhen applied to 1 kilogram of mass, will accelerate themass at a rate of 1 meter per second squared (m/s2).That is: 1N (1kg)(1m/s2).

International System (SI) of Units

British Gravitational (BG) System

• In the British Gravitational (BG) system of units, the unit oflength is a foot (ft), which is equal to 0.3048 meter

• The unit of temperature is expressed in degree Fahrenheit (F)or in terms of absolute temperature degree Rankine (R).

• The relationship between the degree Fahrenheit and degreeRankine is given by:

British Gravitational (BG) System

• The relationship between degree Fahrenheit and degreeCelsius is given by:

• The relationship between the degree Rankine and the Kelvinby:

British Gravitational (BG) System

U.S. Customary Units

• The unit of length is a foot (ft), which is equal to 0.3048meter.

• The unit of mass is a pound mass (lbm), which is equalto 0.453592 kg; and the unit of time is a second (s).

• The units of temperature in the U.S. Customary systemare identical to the BG system

U.S. Customary Units

UNITS AND MEASUREMENTS

53

The comparison of any physical quantity with its standard unit is calledmeasurement.

Physical QuantitiesAll the quantities in terms of which laws of physics are described, and whosemeasurement is necessary are called physical quantities.

Units• A definite amount of a physical quantity is taken as its standard unit.• The standard unit should be easily reproducible, internationally accepted.

Fundamental Units

Those physical quantities which are independent to each other are calledfundamental quantities and their units are called fundamental units.

Source: www.ncerthelp.com

UNITS AND MEASUREMENTS

54

S.No. Fundamental Quantities Fundamental Units Symbol

1. Length metre m

2. Mass kilogram kg

3. Time second S

4. Temperature kelvin kg

5 Electric current ampere A

6 Luminous intensity candela cd

7 Amount of substance mole mol

Supplementary Fundamental Units

Radian and steradian are two supplementary fundamental units. It measures plane angle and

solid angle respectively.

S. No. Supplementary Fundamental Quantities Supplementary Unit Symbol

1 Plane angle radian rad

2 Solid angle steradian Sr

Source: www.ncerthelp.com

UNITS AND MEASUREMENTS

55

Those physical quantities which are derived from fundamental quantities are calledderived quantities and their units are called derived units.

e.g., velocity, acceleration, force, work etc.

Definitions of Fundamental Units

The seven fundamental units of SI have been defined as under.

1. 1 kilogram: A cylindrical prototype mass made of platinum and iridium alloys ofheight 39 mm and diameter 39 mm. It is mass of 5.0188 x 1025 atoms of carbon-12.

2. 1 metre: 1 metre is the distance that contains 1650763.73 wavelength of orange-redlight of Kr-86.

3. 1 second: 1 second is the time in which cesium atom vibrates 9192631770 times inan atomic clock.

Source: www.ncerthelp.com

UNITS AND MEASUREMENTS

56

Systems of UnitsA system of units is the complete set of units, both fundamental and derived, for all kinds of physical quantities. The common system of units which is used in mechanics are given below:

1. CGS System: In this system, the unit of length is centimeter, the unit of mass is gram and the unit of time is second.

2. FPS System: In this system, the unit of length is foot, the unit of mass is pound and the unit of time is second.

3. MKS System: In this system, the unit of length is meter, the unit of mass is kilogram and the unit of time is second.

4. SI System: This system contain seven fundamental units and two supplementary fundamental units.

Source: www.ncerthelp.com

UNITS AND MEASUREMENTS

57

Dimensions:

• Dimensions of any physical quantity are those powers which are raised on fundamental units to express its unit.

• The expression which shows how and which of the base quantities represent the dimensions of a physical quantity, is called the dimensional formula.

S.No . Physical Quantity Dimensional Formula MKS Unit

1 Area [L2] metre2

2 Volume [L3] metre3

3 Velocity [LT-1] ms-1

4 Acceleration [LT-2] ms-2

5 Force [MLT-2] newton (N)

6 Work or energy [ML2T-2] joule (J)

7 Power [ML2T-3] J s-1 or watt

8 Pressure or stress [ML-1T-2] Nm-2

9 Linear momentum or Impulse [MLT-1] kg ms-1

10 Density [ML-3] kg m-3Source: www.ncerthelp.com

UNITS AND MEASUREMENTS

58

S.No . Physical Quantity Dimensional Formula MKS Unit

11 Strain Dimensionless Unitless

12 Modulus of elasticity [ML-1T-2] Nm-2

13 Surface tension [MT-2] Nm-1

14 Velocity gradient T-1 second-1

15 Coefficient of velocity [ML-1T-1] kg m-1s-1

16 Gravitational constant [M-1L3T-2] Nm2/kg2

17 Moment of inertia [ML2] kg m2

18 Angular velocity [T-1] rad/s

19 Angular acceleration [T-2] rad/S2

20 Angular momentum [ML2T-1] kg m2S-1

21 Specific heat L2T-2θ-1 kcal kg-1K-1

22 Latent heat [L2T-2] kcal/kg

23 Planck’s constant ML2T-1 J-s

24 Universal gas constant [ML2T-2θ-1] J/mol-K

UNITS AND MEASUREMENTS

59

Homogeneity PrincipleIf the dimensions of left-hand side of an equation are equal to the dimensions of right hand side of the equation, then the equation is dimensionally correct. This is known as homogeneity principle.

Mathematically [LHS] = [RHS]

Applications of Dimensions1. To check the accuracy of physical equations.2. To change a physical quantity from one system of units to another system of units.3. To obtain a relation between different physical quantities.

Dimensional Analysis

What is Dimensional Analysis?

• Have you ever used a map?

• Since the map is a small-scale representation of a large area, there is a scale that you can use to convert from small-scale units to large-scale units—for example, going from inches to miles or from cm to km.

What is Dimensional Analysis?

Ex: 3 cm = 50 km

What is Dimensional Analysis?

• Have you ever been to a foreign country?

• One of the most important things to do when visiting another country is to exchange currency.

• For example, one United States dollar equals 1509.7 Lebanese Pounds. (9/29/17)

What is Dimensional Analysis?

• Whenever you use a map or exchange currency, you are utilizing the scientific method of dimensional analysis.

• Dimensional analysis is a problem-solving method that uses fractions to cancel units.

• It is used to go from one unit to another.

How Does Dimensional Analysis Work?• A conversion factor, or a fraction that is equal to

one (numerator equals the denominator), is used, along with what you’re given, to determine what the new unit will be.

Using the Conversion Factor

Objective: Make unit conversions

Fill in the Missing Numbers

1 foot = _____ inches

1 meter = _____ centimeters

1 pound = _______ ounces

1 minute = ______ seconds

1 hour = ________ minutes

1 day = __________ seconds

12

100

16

60

60

86,400

Unit conversion factor

A unit conversion factor is a fraction whose

numerator and denominator are equivalent

measures. Some common unit conversion factors

are given below. You can also use the reciprocal of

these.

1 ft

? in.

1 yd

? ft

1 mi

? ft

1lb

? oz

1 pt

? c

1 qt

? pt

1 gal

? qt

1 hr

? min

1 min

? s

1m

? cm

1km

? m

Unit conversion factor

A unit conversion factor is a fraction whose

numerator and denominator are equivalent

measures. Some common unit conversion factors

are given below. You can also use the reciprocal of

these.

Choose a unit conversion factor that…

• Introduces the unit you want in the answer

•Cancels out the original unit so that the one you want is all that is left.

“Canceling” out Words

inches feet

feet miles

seconds hours

minutes seconds

inches

miles=

hours

minutes=

Practice: Choose the appropriate conversion factor.

Inches to feet 1 ft

12 in.

12 in.

1 ft

Minutes to hours60 min

1 hr

1 hr

60 min

Meters to centimeters 1 m

100 cm

100 cm

1 m

inches

1

minutes

1

meters

1

Let’s try some examples together…1. Suppose there are 12 slices of pizza in one pizza.

How many slices are in 7 pizzas?

Given: 7 pizzas

Want: # of slices

Conversion: 12 slices = one pizza

7 pizzas

1

Solution

• Check your work…

X12 slices

1 pizza=

84 slices

Let’s try some examples together…2. How old are you in days?

Given: 17 years

Want: # of days

Conversion: 365 days = one year

Solution

• Check your work…

17 years

1X

365 days

1 year=

6205 days

Let’s try some examples together…3. There are 2.54 cm in one inch. How many inches

are in 17.3 cm?

Given: 17.3 cm

Want: # of inches

Conversion: 2.54 cm = one inch

Solution

• Check your work…

17.3 cm

1X

1 inch

2.54 cm= 6.81 inches

Be careful!!! The fraction bar means divide.

A bucket holds 16 quarts. How many gallons of

water will fill the bucket? Use a unit conversion

factor to convert the units.

What are the two conversion

factors comparing quarts and

gallons?

4 qt

1 gal

1 gal

4 qt

Which one will “cancel”

quarts?

16 qt •16 gal

4 gal4

= =

Now, you try…

1. Determine the number of eggs in 23 dozen eggs.

2. If one package of gum has 10 pieces, how many pieces are in 0.023 packages of gum?

Multiple-Step Problems

• Most problems are not simple one-step solutions. Sometimes, you will have to perform multiple conversions.

• Example: How old are you in hours?

Given: 17 years

Want: # of hours

Conversion #1: 365 days = one year

Conversion #2: 24 hours = one day

Solution

• Check your work…

17 years

1X

365 days

1 yearX

24 hours

1 day=

148,920 hours

Making Rate Conversions

Use a unit conversion to convert the units within each rate

Convert 80 miles per hour to feet per hour.

80 mi

1 hr

5,280 ft

1 mi

80 5280 ft

1 hr

=

422,400 ft

1 hr=

Convert 63,360 feet per hour to miles per hour.

63,360 ft

1 hr

1 mi

5280 ft 63,360 ft 1 mi

1 hr 5280 ft

=

12 mi

1 hr=

You Try it!

Convert 32 feet per second to inches per second.

32 ft 12 in 32 ft 12 in 384 ft =

1 s 1 ft 1 s 1 ft 1 s

=

A craft store charges $1.75 per foot for lace. How much per yard

is this?

$1.75 3 ft $1.75 3 ft $5.25 =

1 ft 1 yd 1 ft 1 yd 1 yd

=

in

Word Problems

The average American eats 23 pounds

of pizza per year. Find the number of

ounces the average American eats per

year.

23 lb 16 oz 23 16 368 oz

1 yr 1 lb 1 1 yr

= =

The average American eats 368 ounces

of pizza per year.

Word Problems…

A car traveled 60 miles on a road in 2 hours.

How many feet per second was the car

traveling? Hint: Set up the words first.

mi

hr

ft

mi

hr

s

How many seconds

in an hour?60 s 60 min 3600 s

1 min 1hr 1 hr =

1 hr

3600 s

60 mi

2 hr

5,280 ft

1 mi

60 5,280

2 3600

= =

44 ft

1 s=

The car traveled 44 feet per second.

Unit Conversion

• Example 6.1 :

A person who is 6 feet and 1 inch tall and weighs185 pound force (lbf) is driving a car at a speed of65 miles per hour over a distance of 25 miles. Theoutside air temperature is 80F and has a density of0.0735 pound mass per cubic foot (lbm/ft3).Convert all of the values given in this example fromU.S. Customary Units to SI units.

Unit Conversion (Example 6.1 )

Unit Conversion (cont. Example 6.1 )

Unit Conversion (Example 6.2 )

Work out Example 6.2 at home .If you have any question askme .

Dimensional Homogeneity

• What do we mean by “dimensionallyhomogeneous?”

Can you, say, add someone’s height who is 6 feet tallto his weight of 185 lbf and his body temperature of98F?! Of course not!

Dimensional Homogeneity (Example 6.3 )

• For Equation 6.1 to be dimensionally homogeneous, the units on the left-hand side of the equation must equal the units on the right-hand side. This equality requires the modulus of elasticity to have the units of N/m2, as follows:

Dimensional Homogeneity (cont. Example 6.3 )

Numerical versus Symbolic Solutions

➢When you take your engineering classes, you need to be aware oftwo important things:

(1) understanding the basic concepts and principles associated withthat class

(2)how to apply them to solve real physical problems (situations)

➢Homework problems in engineering typically require either anumerical or a symbolic solution.

➢ For problems that require numerical solution, data is given. Incontrast, in the symbolic solution, the steps and the final answerare presented with variables that could be substituted with data.

Numerical versus Symbolic Solutions (Example 6.4)

• Determine the load that can be lifted by the hydraulic systemshown. All of the necessary information is shown in the Figure.

Numerical versus Symbolic Solutions (Example 6.4)

➢Numerical Solution:

We start by making use of the given data and substitutingthem into appropriate equations as follows.

Numerical versus Symbolic Solutions (Example 6.4)

➢Symbolic Solution:

For this problem, we could start with the equation that relatesF2 to F1, and then simplify the similar quantities such as p andg in the following manner:

Significant Digits (Figures)

• One half of the smallest scale division commonly is calledthe least count of the measuring instrument.

• For example, referring to Figure 6.4, it should be clear thatthe least count for the thermometer is 1F (the smallestdivision is 2F), for the ruler is 0.05 in., and for the pressuregage is 0.5 inches of water.

• Therefore, using the given thermometer, it would beincorrect to record the air temperature as 71.25F and lateruse this value to carry out other calculations. Instead, itshould be recorded as 71 1F.

• This way, you are telling the reader or the user of yourmeasurement that the temperature reading falls between 70Fand 72F.

Significant Digits (Figures)

• Significant digits are numbers zero through nine.However, when zeros are used to show the position of adecimal point, they are not considered significant digits.

• For example, each of the following numbers 175, 25.5,1.85, and 0.00125 has three significant digits. Note thezeros in number 0.00125 are not considered as significantdigits, since they are used to show the position of thedecimal point

Significant Digits (Figures)

• The number of significant digits for the number 1500 is notclear. It could be interpreted as having two, three, or foursignificant digits based on what the role of the zeros is.

• In this case, if the number 1500 was expressed by 1.5 *10^3,15*10^2, or 0.015 *10^5, it would be clear that it has twosignificant digits. By expressing the number using the powerof ten, we can make its accuracy more clear.

• However, if the number was initially expressed as 1500.0, thenit has four significant digits and would imply that the accuracyof the number is known to 1/10000.

Significant Digits (Figures)

➢Addition and Subtraction Rules

➢Multiplication and Division Rules

107

Thank You

108