1. introduction and fundamental concepts [1]
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1. Introduction and Fundamental Concepts [1] . System-Surroundings-Interaction Physical Phenomena, Physical Quantities, and Physical Relations Physical Quantity Dimensions Numerical value Unit of Measure Dimensions – A simple key to some physical understanding of fluid mechanics - PowerPoint PPT PresentationTRANSCRIPT
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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]
)(][,2)(][,2
.1.3.2LengthLlml
LengthLlml
DimensionQmeasureofunitQmeasureofunitwrtvaluenumerical
][q
q
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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
)(][,2.1.3.2
LengthLlmlDimensionQmeasureofunitQunitwrtvaluenumerical
Physical QuantityDescribing A Physical Quantity
][q
Q
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