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1. System-Surroundings-Interaction

2. Physical Phenomena, Physical Quantities, and Physical Relations

3. Physical Quantity

1. Dimensions

2. Numerical value

3. Unit of Measure

Dimensions – A simple key to some physical understanding of fluid mechanics

Systems of Dimensions and Units

4. Physical Quantity

Units of Measure and Principle of Absolute Significance of Relative Magnitude (PASRM)

Dimension is a power-law monomial

5. Physical Relation and Principle of Dimensional Homogeneity

6. Dimensionless Variables and ‘Measuring/Scaling’

1. Introduction and Fundamental Concepts [1]

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(Physical) System

Universe / Isolated System Surroundings

Interaction Mechanical interaction (force)

Thermal interaction (energy and energy transfer)

Electrical, Chemical, etc.

Fundamental Concept: System-Surroundings-Interaction

The very first task in any one problem:

Identify the system

Identify the surroundings

Identify the interactions between the system and its surroundings, e.g., Mechanics - Force (identify all the forces on the system by its surroundings)

Thermodynamics - Energy and Energy Transfer (identify all forms of energy and energy transfer between the system and its

surroundings)

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Example: Thermodynamics - Heating Water

Water 1 liter in a container in atmosphere.

Add heat of the amount Q. [Assume no heat loss elsewhere.]

QUESTION: How much is the temperature rise?

?

?

mcQT

TmcQ

Q

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Energy transfer as heat Q into the two systems are not the same.

?

?

mcQT

TmcQ

Q

System 1: Water + Container

• How much is the energy transfer as heat into the System 1?

?1SystemmcQT

?Q

System 2: Water only

• How much is the energy transfer as heat into the System 2?

?2SystemmcQT

?Q

Two different systems have two different (energy) interactions with their own surroundings.

Obviously, at least the mass of the two systems are not the same.

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Example: (Fluid) Mechanics - Flow in Pipe

System 1:

Water stream in the pipe only, exclude the solid pipe and flange.

System 2:

Water stream in the pipe, and the solid pipe and flange (cut through the bolts).

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External forces on the two systems are not the same.

System 2:

• There are forces at the solid bolts acting on the system.

F

Two different systems have two different (mechanical) interactions with their own surroundings.

Obviously, at least the mass of the two systems are not the same.

System 1:

• Pressure and shear stress distributions on the surfaces only.• No force at the solid bolt.

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Key Point: Define your system first before you apply an equation.

Since the application of basic principles / equation is always to a

specific system,

define your system first before you apply an equation.

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Classification of Systems

Interaction between system and surroundings

Exchange of Mass (between system and surroundings)

Exchange of Energy (between system and surroundings)

Isolated system No No

Closed system(Identified mass, Control mass, Material volume)

No Yes

Open system(Identified volume, Control volume)

Yes Yes

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Example: Various types of systems

Isolated system: Insulated hot water bottle

(approximately isolated over a short period of time, no energy absorption

due to radiation, etc.)

Closed system Open system(part of the mass is

evaporated out of the system)

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physical phenomena

physical quantities

physical relations [relations among physical quantities]

Boeing 747-400

Cruising speed Mach Number = 0.85 (Compressible Flows).

(From http://www.boeing.com/companyoffices/gallery/images/commercial/747400-06.html)

Physical Phenomena, Physical Quantities, and Physical Relations

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Physical quantity is a concept.

A quantifiable/measurable attribute we assign to a particular characteristic of nature that we

observe.

We must find a way to ‘quantify’ it.

1. Describing a physical quantity. We need 3 things:

1. Dimension

2. Numerical value with respect to the unit of measure

3. Unit of measure

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Q

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2. Q and Q must go together.

• Change the unit of measure Q, the numerical value Q must be changed

accordingly.

angstrom

mm

cm

in

ftyardmile

milenautical

unitalastronomicau

LengthLlmlDimensionQmeasureofunitQunitwrtvaluenumerical

10

3

2

1

3

3

11

.1.3.2

102

102

102

1087.7

56.619.2

1024.1

1008.1

)(1034.1

,][,2

In fact, we can change these numerical values to any numerical value that we want so long as we choose the corresponding unit of measure Q.

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Key Point: Q and Q must go together.

Always write the corresponding unit Q for the corresponding

numerical value Q of a physical quantity.

[Except when that quantity is dimensionless.]

m = 55 what? 5 kg

5 lbm5 ton?

m = 5 ton

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Fundamental Concept: Quantification and Measure(ment)

(Any sort of) Quantification is always based on measure, unit of

measure, measurement.

There is a degree of arbitrariness in choosing a unit of measure.

angstrom

mm

cm

in

ftyardmile

milenautical

unitalastronomicau

LengthLlmlDimensionQmeasureofunitQunitwrtvaluenumerical

10

3

2

1

3

3

11

.1.3.2

102

102

102

1087.7

56.619.2

1024.1

1008.1

)(1034.1

,][,2

In fact, we can change these numerical values to any numerical value that we want so long as we choose the corresponding unit of measure Q.

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The followings cannot be emphasized enough.

To gain some physical understanding of fluid (mechanics), pay

attention to the dimensions of the physical quantity/relation of interest.

Choose any dimension that you can relate physically, not necessarily –

and often not - MLtT.

Key Point: Dimensions - A simple key to gain some understanding of fluid mechanics (or rather physics in general)

• Enthalpy h has the dimension of • L2t-2 “What is this?”• Energy/Mass O.k. This, I can relate.

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More Example: Dimensions - A simple key to gain some understanding of fluid mechanics (or rather physics in general)

Specific heat C has the dimension of

• L2t-2T-1 “What is this?”

• O.k. This, I can relate.

Note reads “Energy per unit mass per unit (change in) temp”

• I can guess that C should somehow be related to the amount of energy per unit mass per unit (change in) temperature.

TempMassEnergy

TempMassEnergy

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Systems of Dimensions and Units

1. Primary Quantities and Primary Dimensions

Choose a set of primary quantities (and consider their dimensions to be

independent).

Three systems of common use are MLtT, FLtT, FMLtT

2. Units of Measure for Each Primary Quantity

Choose a unit of measure for each primary quantity.

MLtT: In SI: M – kg, L – m, t – sec, T – K

3. Derived Quantities/Dimensions

Through physical relations, we have derived quantities and their dimensions

e.g., [Velocity] = L/t, [Acceleration] = L/t2 [Force] = ML/t2, etc.

][][][

trV

dtrdV

By definition

2]][[][ MLtamFamF By law

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Key Point: There is some arbitrariness in choosing primary quantities and dimensions.

Conceptually, for example, we can choose

ELtT Energy-Length-Time-Temp

as a set of primary quantities and primary dimensions in place of

MLtT

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Example of Systems of Units: SI

From Physics Laboratory, The National Institute of Standards and Technology (NIST)’s web page: http://physics.nist.gov/cuu/Units/SIdiagram2.html

Note that there can be some characters missing from this diagram due to font and file related issues during the making of the presentation slide. Go to the NIST’s web page given above for the original.

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Fundamental Concepts: Physical Quantity: Chosen System of Units and The Principle of Absolute Significance of Relative Magnitude (PASRM) To have some physical sense, (we prefer to use and/or) we require (of systems of units to be

used) that

the ratio of magnitudes of any two concrete physical quantities should not depend on the system of units used.

rA

A

A

A

2

1

22

21

22

ftin

ftin

min

min A1

A2

The ratio A2/A1 should be the same regardless of whether the numerical values of A1 and A2 are

expressed in m2 or ft2.

???Cin

Cin Kin Kin

o1

o2

1

2

T

TTTWhat about

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Dimension (function) is a power-law monomial

It cannot be, e.g., MLeqMq ][),sin(][

In other words, the argument of these functions must be dimensionless, e.g.,

if we know that

we then know that

)sin()( batVtV o

1][,/1][1][ 0000 btaTtLMbat

dcba TtLMq ][11][ tLV

The dimensions of a derived quantity q must be in the form of a power-law monomial

e.g.,

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Dimensionless Quantity

1][ 0000 TtLMq

Dimensionless quantity q has the dimension of

Example

1][: 0000 TtLMrs Angle (radian)

1][//: 0000 TtLM

InputEnergyWorkoutputEnergyWork Efficiency

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Independent Dimensions

1

][][][ Lt

trV

dtrdV

By definition

2]][[][ MLtamFamF By law

In MLtT system, since the dimension of velocity V can be written as the power-law monomial of L and t,

V = L1t-1

the physical quantities (Velocity, Length, time) do not have independent dimension.

Independent Dimensions: Physical quantities q1,…, qr are said to have

independent dimensions

if none of these quantities has a dimension function that can be written in terms of a power-law

monomial of the dimensions of the remaining quantities. (Barenblatt, 1996)

Similarly, (Length, Time, Mass, and Force) do not have independent dimension since, according to Newton’s

Second law of motion,

F = MLt-2

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Requirement/Premise: Any equation that describes a physical relation cannot be dependent upon an arbitrary

choice of units (within a given class of systems of units).

Principle of Dimensional Homogeneity (PDH):

All physically meaningful equations, i.e., physical relation/equation, are dimensionally

homogeneous. (Smits, 2000)

If is a physical equation,

then have the same dimension,

i.e.,

Useful for checking our derived result: [We shall deal only with physical equations.]

Physical Relation and Principle of Dimensional Homogeneity (PDH)

Physical Relation Dimensionally homogeneous

~ Physical Relation(We derive something wrong somewhere.)

~ Dimensionally homogeneous

21 XXY

21 XXY

,,, 21 XXY

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Example: The Use of The Principle of Dimensional Homogeneity in Checking Our Results

]41[][][

?41

2

2

gtuts

gtuts

However, we cannot use PDH to tell whether something is definitely correct.

VelocitytLLLthg

VelocityV

12/32/12 )(]2[

][

The result is not correct.

We can use PDH to tell whether something is definitely wrong.

?2 hgV

QUESTION: Without the knowledge of mechanics, can you tell whether this result/equation is

wrong:

Even though our result is dimensionally homogeneous, we cannot tell whether it is correct by

PDH alone.

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Dimensionless Variables and ‘Measuring/Scaling’ Dimensionless Variables:

From

if we divide through by one of the term, say Xn, we obtain

The new variables, e.g.,

then have no dimension. We call these variables dimensionless variables.

nXXXY 21

1121

n

n

nnn XX

XX

XX

XY

n

ii X

XZ

Scaling/Measuring:

Xi is Zi fraction of Xn.

Scaling:

One can think of the above process as the measurement/scaling of the variables Y, X1,…,Xn-1, with Xn.

In other words, we measure, e.g., Xi as a fraction (or per cent) of Xn, or we measure Xi relative to Xn

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Example: Dimensionless Variables and ‘Measuring/Scaling’

D(= 10 cm)

L Unit of measure

cm

D

3001

cmL

The pipe is 300 cm long - unit of measure = cm

30DL

The pipe is 30 D long - unit of measure = D

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Key Idea: Use the units/scales of measure that are inherent in the problem itself, not the man-made one irrelevant to the problem.

D(= 10 cm)

L Unit of measure

cm

D

3001

cmL

30DL

• In order to understand physical phenomena better, we prefer to use the

units/scales of measure that exist in the problem itself, not the man-made

one irrelevant to the problem.

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While the numerical values of power output in the units of Watt and hp are not the same,

2,000 W VS 2,000/746 = 2.68 hp

the numerical value of the dimensionless variables efficiency is the same regardless of whether

we use W or hp:

Other examples of dimensionless variables are

Reynolds number Re

Mach number M

Key Point: The numerical value of dimensionless variable does not depend on the (appropriate) system of units used.

5.0746/000,4746/000,2

000,4000,2

1][//: 0000

hphp

WW

TtLMInputEnergyWork

outputEnergyWork