Transcript Slide 1

1. Introduction and Fundamental Concepts [1]
1.
System-Surroundings-Interaction
2.
Physical Phenomena, Physical Quantities, and Physical Relations
3.
Physical Quantity
q
[q ]
1.
Dimensions
2.
Numerical value
Q
3.
Unit of Measure
Q
l  2 m,
l
4.
[l ]  L ( Length)
2
m

2. numerical value wrt unit of measure Q
3. unit of measure Q
[
l]
 L ( Length)
1. Dimension

Dimensions – A simple key to some physical understanding of fluid mechanics

Systems of Dimensions and Units
Physical Quantity

Units of Measure

Dimension is a power-law monomial
and
and
Principle of Absolute Significance of Relative Magnitude (PASRM)
5.
Physical Relation
6.
Dimensionless Variables and ‘Measuring/Scaling’
abj
,
Principle of Dimensional Homogeneity
1
Fundamental Concept: System-Surroundings-Interaction
Universe / Isolated System
Surroundings
Interaction
 Mechanical interaction (force)
 Thermal interaction (energy and energy transfer)
(Physical) System
 Electrical, Chemical, etc.
 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.,
abj
 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)
2
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?
Q
abj
Q  mcT
?
 T 
Q
?
mc
3
Energy transfer as heat Q into the two systems are not the same.
Two different systems have two different (energy) interactions with their own surroundings.
System 1:
Q  mcT
?
 T 
Q
?
mc
Water + Container
Q


T

System 1 ?


mc 

• How much is the energy transfer
as heat into the System 1?
Q?
Obviously, at least the mass of the two systems are not the same.
System 2: Water only
Q 


T

System 2 ?


mc 

Q
abj
Q?
• How much is the energy transfer as
heat into the System 2?
4
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).
abj
5
External forces on the two systems are not the same.
Two different systems have two different (mechanical) interactions with their own surroundings.
System 1:
• Pressure and shear stress distributions on the
surfaces only.
• No force at the solid bolt.
Obviously, at least the mass of the two systems are not the same.
F
System 2:
• There are forces at the solid bolts acting on the system.
abj
6
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.
abj
7
Classification of Systems
Interaction between system and surroundings
Exchange of Mass
Exchange of Energy
(between system and surroundings)
(between system and surroundings)
Isolated system
No
No
Closed system
No
Yes
Yes
Yes
(Identified mass, Control mass,
Material volume)
Open system
(Identified volume, Control volume)
abj
8
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.)
abj
Closed system
Open system
(part of the mass is
evaporated out of the
system)
9
Physical Phenomena, Physical Quantities, and Physical Relations

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)
abj
10
Physical Quantity
Describing A Physical Quantity

Physical quantity is a concept.

A quantifiable/measurable attribute we assign to a particular characteristic of nature that we
observe.

l
1.
abj
We must find a way to ‘quantify’ it.
2
m

2. numerical value wrt unit Q
3. unit of measure Q
Describing a physical quantity.
,
[
l]
 L ( Length)
1. Dimension
We need 3 things:
1.
Dimension
[q ]
2.
Numerical value with respect to the unit of measure
Q
3.
Unit of measure
Q
11
2.
Q and Q must go together.
•
Change the unit of measure Q, the numerical value Q must be changed
accordingly.
l
abj

2
m

,
2. numerical value wrt unit Q
3. unit of measure Q
[
l]
 L, Length
1. Dimension

1.34 1011
au (astronomic
al unit)

1.08 103
nauticalmile

1.24 103
mile


2.19
6.56
yard
ft

7.87  101
in

2  102
cm

2  103
mm

2  1010
angstrom
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.
12
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=5
5 what?
5 kg
5 lbm
5 ton?
m = 5 ton
abj
13
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.
l
abj

2
m

,
2. numerical value wrt unit Q
3. unit of measure Q
[
l]
 L, Length
1. Dimension

1.34 1011
au (astronomic
al unit)

1.08 103
nauticalmile

1.24 103
mile


2.19
6.56
yard
ft

7.87  101
in

2  102
cm

2  103
mm

2  1010
angstrom
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.
14
Key Point:

Dimensions - A simple key to gain some understanding of fluid
mechanics (or rather physics in general)
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.
• Enthalpy
abj
h has the dimension of
• L2t-2
“What is this?”
• Energy/Mass
O.k. This, I can relate.
15
More Example:
Dimensions - A simple key to gain some understanding of fluid mechanics (or rather physics in general)
 Specific heat
Note
•
L2t-2T-1
•
Energy
Mass  Temp
Energy
Mass  Temp
C
has the dimension of
“What is this?”
O.k. This, I can relate.
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.
abj
16
Systems of Dimensions and Units
1.
Primary Quantities and Primary Dimensions

Choose a set of primary quantities (and consider their dimensions to be
independent).

2.
Three systems of common use are
FMLtT
Choose a unit of measure for each primary quantity.
MLtT:
In SI:
M – kg,
L – m,
t – sec,
T–K
Derived Quantities/Dimensions

Through physical relations, we have derived quantities and their dimensions
By definition
By law

abj
FLtT,
Units of Measure for Each Primary Quantity

3.
MLtT,
e.g.,
 dr
V 
dt


F  ma
[Velocity] = L/t,




[r ]
[V ] 
[t ]


[ F ]  [m][a ]  MLt 2
[Acceleration] = L/t2
[Force] = ML/t2, etc.
17
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
abj
18
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
abj
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.
19
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.
A2
A1
 A2 in m 2

 A in m 2
 1
  A2 in ft 2

  A in ft 2
  1

r


The ratio A2/A1 should be the same regardless of whether the numerical values of A1 and A2 are
expressed in m2 or ft2.
What about
abj
 T2 in K   T2 in o C 
 ???

  
o


 T1 in K   T1 in C 
20
Dimension (function) is a power-law monomial
 The dimensions of a derived quantity q must be in the form of a power-law monomial
[q]  M a Lbt cT d
e.g.,

[V ]  L1t 1
 It cannot be, e.g.,
[q]  sin(M ),
[q]  e ML
 In other words, the argument of these functions must be dimensionless, e.g.,
if we know that
we then know that
abj
V (t )  Vo sin(at  b)
[at  b]  M 0 L0t 0T 0  1

[ a]  1 / t ,
[b]  1
21
Dimensionless Quantity
 Dimensionless quantity q has the dimension of
[q]  M 0 L0t 0T 0  1
Example
abj
Efficiency
 :
W ork/ Energy output
W ork/ Energy Input
Angle (radian)
 :
s
r


[ ]  M 0 L0 t 0T 0  1
[ ]  M 0 L0t 0T 0  1
22
Independent Dimensions
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)
By definition
By law
 dr
V 
dt


F  ma




[r ]
[V ] 
 Lt 1
[t ]


[ F ]  [m][a ]  MLt 2
 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.
 Similarly, (Length, Time, Mass, and Force) do not have independent dimension since, according to Newton’s Second
law of motion,
abj
F = MLt-2
23
Physical Relation

and Principle of Dimensional Homogeneity (PDH)
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
then
Y  X1  X 2  
Y , X 1 , X 2 ,
is a physical equation,
have the same dimension, i.e.,
Y   X1  X 2   
Useful for checking our derived result: [We shall deal only with physical equations.]
abj
Physical Relation
Dimensionally homogeneous
~ Dimensionally homogeneous
~ Physical Relation
(We derive something wrong somewhere.)
24
Example: The Use of The Principle of Dimensional Homogeneity in Checking
Our Results
 We can use PDH to tell whether something is definitely wrong.
 QUESTION:
Without the knowledge of mechanics, can you tell whether this result/equation is
V  2g  h
wrong:
?
[V ]  Velocity
[ 2 g  h]  ( Lt  2 )1/ 2  L

L3 / 2t 1

Velocity
The result is not correct.
 However, we cannot use PDH to tell whether something is definitely correct.
Even though our result is dimensionally homogeneous, we cannot tell whether it is correct by
PDH alone.
1 2
gt
?
4
1
[ s]  [ut]  [ gt 2 ]
4
s  ut 
abj
25
Dimensionless Variables and ‘Measuring/Scaling’
 Dimensionless Variables:
From
Y  X1  X 2    X n
if we divide through by one of the term, say Xn, we obtain
 Y   X1   X 2 
X




     n 1   1
X  X  X 
 X 
n 
 n  n  n

The new variables, e.g.,
X 
Z i   i 
 Xn 
Scaling/Measuring:
Xi is Zi fraction of Xn.
then have no dimension. We call these variables dimensionless variables.

abj
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
26
Example:
Dimensionless Variables and ‘Measuring/Scaling’
Unit of measure
L
cm
D
D
(= 10 cm)
 The pipe is
300 cm
long - unit of measure = cm
L
 300
1 cm
 The pipe is
30
D
long - unit of measure = D
L
 30
D
abj
27
Key Idea:
Use the units/scales of measure that are inherent in the
problem itself, not the man-made one irrelevant to the problem.
Unit of measure
L
cm
D
D
(= 10 cm)
L
 300
1 cm
L
 30
D
• 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.
abj
28
Key Point:

The numerical value of dimensionless variable does not
depend on the (appropriate) system of units used.
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:
 :

W ork/ Energy output
W ork/ Energy Input

[ ]  M 0 L0t 0T 0  1
2,000 W 2,000/ 746 hp

4,000 W 4,000/ 746 hp
 0.5
Other examples of dimensionless variables are
abj

Reynolds number Re

Mach number M
29