Transcript - Mitra.ac.in

Experimental Modeling
-Approach
1
What is model?
• Models are set of variables and their
interrelationships designed to represent
some real system in part or whole.
• A body of information about a system
gathered for the purpose of studying the
system.
• Research plays a major role in model
building and decision making process.
Example of a model
Y = 63.5 x A0.37 x B0.63 x C-1.85 x D3.48x E6.76
Where
• Y
= dependent variable,
• A,B,C,D,&E =Independent variables
• Values of the indices(0.37, 0.63, -1.85, 6.76) and const.(63.5)
are computed through experimentation
Objectives of model formulation
•
Prediction of future
•
Aid for decision making
•
Act as input for optimization of the variables
•
As an tool for Improvement of productivity/
effectiveness
Theoretical Approach
Apply basic laws of Mechanics &
Physics (Force Balance, Momentum
Balance, Energy Balance, Mass
Balance)
Form Mathematical
model of process
Verify
Experimentally
5
Experimental Approach
Vary all the INPUTS, of the process over a
widest possible range
Collect experimental data of
process OUTPUTS .
Obtain empirical relationship between inputs
and the outputs of the process by analyzing
the data.
Optimize established
relations.
6
STEPS IN EXPERIMENTATL APPROACH
•
IDENTIFICATION OF PROCESS VARIABLES
•
DIMENSIONAL ANALYSIS /REDUCTION OF VARIABLES
•
TEST PLANNING (TEST ENVELOPE/
•
DESIGN OF EXPERIMENTAL SET UP
•
DATA PURIFICATION
•
FORMULATION OF MODEL
POINTS /
SEQUENCE / PLAN)
7
Dimensional Analysis
• Used primarily as experimental tool to combine
many experimental variables into one in the
fluid mechanics and heat transfer.
• Now a days is frequently presented as a
pedagogical device.
• The main is purpose of making experimentation
shorter without the loss of control.
8
Dimensions /Quantities
1. Fundamental Dimensions(Primary Quantities)
•
•
•
•
Mass (M)
Length (L)
Time (T)
Temperature (θ)
2. Derived Dimensions (Secondary Quantities)
Quantities, which possess more than one
fundamental dimensions.
9
Types of the derived quantities
S.N
Category
Distinguishing feature
Example
1
Dimensional
Have dimensions but
no physical value
Force, Velocity,
Power
Specific gravity,
strain angle
constants
Neither have
dimensions nor fixed
value
Have fixed dimensions
and fixed value
Dimensionles
s constants
Have no dimensions
but have fixed values
1, 2, 3, Pi, e
variables
2
Dimensionles
s variables
3
Dimensional
4
Grav.constant,
Planck’s Const.
10
Dimensional formulae of the derived quantities
Sr.
No.
1
2
Physical
Quantity
Relation with other
physical quantities
Area
Length x Breadth
Volume
Length x Breadth x
Height
3
Density
Mass/Volume
4
Speed or Vel.
Distance/Time
5
Acceleration
Velocity/Time
6
Force
Mass x Acceleration
7
Momentum
Mass x Velocity
8
Work
Force x Distance
Dimensional
Formula
[L]x[L]=[L2]
[L]x[L]x[L]=[L3]
[M]/[L3]=ML-3T0]
[L]/[T]=M0LT-1]
[M0LT1]/[T]=[M0LT-2]
[M]x[MILT-2] =
[MLT-2]
[M]x[M0LT-1] =
[MLT-1]
[MLT-2]x[L] =
[ML2T-2]
SI
Unit
m2
m3
kg m-3
ms-1
ms-2
N
kg m
s-1
Nm
11
9
Power
Work/Time
10
Pressure
Force/Area
11
12
Kinetic
energy
Potential
energy
13
Impulse
14
Torque
15
Stress
16
Strain
17
Elasticity
18
Surface
tension
½ x Mass x
(Velocity)2
Mass x g x Distance
Force x time
Force x Distance
Force/Area
Ext. in length
/Original length
Stress/Strain
Force/Length
[ML2T2]/[T]=[ML2T-3]
[MLT2]/[L2]=[ML-1T2]
[M]x[M0LT -1]2
=ML2T-2
[M]X[M0LT2]x[L]=ML2T-2
[MLT-2]x[T]=[
MLT-1]
[MLT2]x[L]=ML2T-2
[MLT2]/[L2]=[ML-1T-2]
[L]/[L]=[M0L0T0
]
[ML-1T-2]
M0L0T0]
= [ML-1T-2]
[MLT2]/[L]=[ML0T-2]
W
Nm-2
Nm
Nm
Ns
Nm
Nm-2
Num
ber
Nm-2
Nm-1
12
Dimensional Equation
v=u+at
Where,
u = initial velocity of the body
a = acceleration
t = the time taken to attain the final velocity v
Writing the formula in the dimensional equation form,
[M0 L T-1] =[M0 L T-1 ] +[M0 L T-2] [M0 L0 T]
According to Principle of homogeneity of dimensions , a physical
relation is dimensionally correct if the dimensions of the fundamental
quantities are same in each and every term on either side of the equation.
This can be seen that all the terms of this equation are having the
dimensions [M0 L T-1].
13
Uses of dimensional equations
•
To check the correctness of a dim.
Equation.
• To recapitulate a forgotten formula.
•
•
•
To derive relation between different
physical quantities.
To convert one system of units to another.
To find dimensions of constants in a given
eqn.
14
Dimensional Analysis applied to Experimentation
After identifying the physical quantities
encountered in the process by study of related
literature or by knowing the governing equation
dimensional analysis can be applied . It helps in
(i) Confirming the correct account of the
physical quantities involved in the process
(ii) Grouping various physical quantities to
make the experimentation compact, economical,
less time taking.
(iii) Proper selection of group of physical
quantities to take care of instrument inaccuracies.
15
Types of Variables
(1) Independent / Fundamental Variables: Variable
that influences the test and can be changed
independently.
(2) Extraneous Variables: Variables that influences
the process but cannot be changed at our will viz.
effect of humidity, human effects.
(3) Controlled Variables: Fundamental variables but
not altered due to practical reasons e.g. acceleration
due to gravity.
(4) Dependent variables / Response variables :
which change due to the change in the values of the
independent variables.
16
Rayleigh’s Method
Let X be an independent variable, which depends
on x1, x2, x3 etc. Then according to Rayleigh's
method, X is a function of x1 , x2 ,x3 - etc. and
mathematically it can be written as
X = f (x1, x2, x3) or
X = k xa1 , xb2 ,xc3
where k is a constant and a, b and c are indices
The values of a, b and c are obtained by comparing
the powers of the fundamental dimension on both
sides.
17
Example:
The drag D experienced by a ship having a
characteristic length l and moving with a speed v
depends upon the fluid mass density ; viscosity 
and acceleration due to gravity g, obtain an
expression for drag in terms of dimensionless
parameters.
Solution: The functional relationship between the
dependent and independent variables can be
expressed as
D=f(L,v,,,g) = K (l)a(v)b()cdge ----(1)
D=pl2v2f[Re,Fr]
18
( MLT 2 )  ( L) a ( LT 1 )b ( ML3 )c ( ML1T 1 ) d ( LT 2 )e
MLT 2  M c d La b3c d eT bd 2e
Equating the powers on both sides and solving in terms of d & e
M : c  d  1  c  1 d
- (2)
L : a  b  3c  d  e  1
- (3)
T : b  d  2e  2  b  2  d  2e
- (4)
From..equation..(3)
a  (2  d  2e)  3(1  d )( d )( d )  e  1
a  2  d  2e  3  3d (d )  e  1
a 1  d  e  1
a  2ed
- (5)
Substituting.. value of a,.b,. & .c.in. terms of d & e in EQ .(1)
D  k (l ) a (v)b (  ) c ( g )e  k (l) 2ed (v) 2d 2e (  )l d  d g e

 e



d



2






lg

vl
v





2
2
2
2







k (l v  )  
k (l v  )   
 lv   2
  
 lg 

 v






 d 








e
19
Buckingham pi (π) Theorem Method
If there are m primary dimensions involved in the n
variables controlling a physical phenomenon, then the
phenomenon can be described by ( n-m ) independent
dimensionless groups.
The repeating variables should be chosen in such a way
that one variable contain geometric property, other
variable contain flow property and third variable contain
fluid property.
If the number of variables are more than the number of
fundamental dimensions (M, L, T) then Buckingham's Pi
theorem becomes less laborious than Rayleigh's method.
20
Example: The testing force F of a supersonic plane
during flight can be considered as dependent upon the
length of the aircraft l, velocity v, air viscosity, air
density, and bulk modulus of air K. Express the
functional relationship between these variables
Solution:
F  f (l , v,  ,  , k )orf ( F , l , v,  ,  , k )  0
-(1)
There are six variables and three primary dimensions. As
n=6, m= 3, n-m=3 dimensionless pi terms will be formed.
Selecting l (geometric property), v( motion property), and
 (fluid property) as repeating variables in each of the
dimensionless group i. e. pi term.
21
 1  la1v b1 c1F
 ( L) a ( LT 1 )b ( ML3 ) c ( MLT 2) )
V  LT 1
  ML3
F  MLT
2
M 0 L0T 0  M c1b13c11T b12
Equating the powers of L, M, T, on either side of this eqn
22
For M
c1+1=0
c1= -1
For L
a1+b1-3c1=1=0
a1= -2
For T
-b1-2=0
b2= -2
1  l

2 2
v
F
1
 F
l v 
2 2
- (2)
2  l v  
a2 b2
c2
 ( L) a2 ( LT 1 )b2 ( ML3 ) c2 ( ML1T 1 )
M LT M
0 0
0
c2 1 a2b2 3 c 21
L
T
b 21
23
Similarly equating the powers of L, M, & T In the above we get

 l v   
lv
2 1
1
1
- (3)
3 l v  k
a3
b3
 ( L)
a3 2
c3
1
0
0
3
c3
1
( LT ) ( ML ) ( ML T
M LT M
0
b3
c3 1
a 3 2 b3 2  3 c 2 1
L
T
2
 b 3 2
For M
c3+1=0
c3= -1
For L
a3+b3-3c3-1=0
a3= 0
For T
-b3-2=0
b3=-2
24
)
 3 
K
 v2
- (4)
Thus the functional relationship will be
 F
 k 
f [ 1 ,  2 ,  3 ]  0  f  2 2 , , 2   0
 l v  vl v 
  k 
2 2
F  l v  
, 2
 v l v 
25
Test Planning
1 Determination of the test envelope
2 Determination of the test points.
3 Determination of the test sequence
4 Determination of the experimental plan
26
1. Determination of Test Envelope
Ideally from -  to + 
Wider range – Higher cost, Unmanageable data
More chances of error, Limitation of Test App.
27
2. Determination of Test points
* More spacing of test points – inaccuracy
3
* Less spacing of test points – expensive
Criteria for selection Test points
a) The relative accuracy of data in different regions of
test envelope.
Many tests show unequal precision over entire test envelope.
If for achieving a precision of P the number of the readings
required are x, then for precision of n times p the number of
readings required are n2 p.
If the precision is doubtful more than normal points may be taken
in this area. If precision is better two times in one region, then
twice the number of observation in that area should be taken for
region with less precision.
28
b) The nature of experimental function (for approx.known exptl. Fn)
Let Pressure drop ∆P = [ K ρ V2] /2g ,
Where ρ = Density V= Velocity of flow &
we want to establish the value of k.
(i) For equal spacing of V at ∆V
The spacing is deficient because at high velocity, the points are
insufficient while at low velocity points are overloaded.
29
ii) For equal spacing of ∆P
We get more points for larger values of ∆P i.e. larger values of
velocity while precision is not good for lower values of velocity.
30
(iii) For uniform & constant precision
We should have points spaced along equal distances or arc length of
experimental curve.
31
3. Test sequence
Basically all test are irreversible for which sequential plan
is adopted. Test are irreversible due to effects of hysterisis
in iron-core, Metallurgical changes in specimen. But
wherever possible in engineering experimentation
randomized plans are used partially or completely.
Random plans are preferred because the observation of
sequential plans shows certain trends due to following
reasons.
i.Natural effects may show a general trend during a test
series.
e.g. effect of humidity & temp on barometric pressure.
ii.Human activities may show a trend during test series
due to increasing skill or increasing boredom.
iii. Mechanical defects may produce a trend with regular
32
variation.
Sequential plans are preferred when
a. Length, cost or difficulties in performing the
experimentation do not permit use of randomized plan.
b. Certain peculiarity or characteristics can to be
shown only by taking data at some regular sequence. 33
Graeco –Latin square
For a machining operation,
Extraneous variables are skill of machinist, his
physical strength, shift of working etc.,
Speeds available are 1,2,3, & 4,
Machines available are w, x, y, & z.
Men operating the machines are A, B, C, & D
Graeco Latin square suggests best randomized
plan to get the optimized results.
34
Every man should work on every machine and on every day. Every
machine should be operated every day and on every cutting speed.
Speeds available are 1,2,3, & 4,
Machines available are w, x, y, & z.
Men operating the machines are A, B, C, & D
Man/Day
A
B
C
D
Mon
Tue
Wed
Thus
1w
2x
3y
4z
3x
4w
1z
2y
2z
1y
4x
3w
4y
3z
2w
1x
35
The concept of randomized block can be used to decide the
optimum m/c speed for a new cutting tool.
R=f(X).
R = Production rate,
X = m/c speed
Rejection percentage not to be exceeded
Extraneous variables, machinist, shift day, machine
Man
M
Shift Day
T
W
A
1
2
3
4
B
1
2
3
4
C
1
2
3
4
D
1
2
3
4
TH
36
Enthusiasm, interest, or dismay generated by a new tool on
first day may gradually fall. Leaving may occur and
production may go-up.
Thus extraneous variable shift day is established –
Randomize the shift day
One solution of randomizing the shift day but not correct
is given below
Man
Shift Day
A
M
4
T
2
W
1
TH
3
B
2
3
1
4
C
3
2
1
4
D
1
3
4
2
37
Decrease in interest at the end of test might suggest a
peaking in the middle speed range. Which is not actually a
speed dependent effect
Correct solution for randomizing the machinist shift day
is given below.
M
Shift Day
T
W
A
1
2
3
4
B
3
4
1
2
C
2
1
4
3
D
4
3
2
1
Man
TH
38
Conditions for proper randomization
1. Each speed must appear every day 2. No man will
execute one speed . Correct solution for randomizing the
machinist shift day
Man
Shift Day
M
T
W
TH
A
1W
2X
3Z
4Y
B
3X
4W
1Y
2Z
C
2Y
1Z
4X
3W
D
4Z
3Y
2W
1X
Machines may have large difference amongst them selves-. To
balance tent 4 m/c – W, X, Y, Z
Condition – each m/c to be used by each operator for each speed &
39
every day.
Case study of Seat design
1 Identification of variables or quantities
(A) Dependent or the response variables



Productivity
Human energy
Electrical energy
(B) Independent variables




Geometric features/variables of seat.
Anthropometric features/variables of operator.
Environmental variables.
Other variables.
40
41
Dimensional analysis
1 Pi-term

1
relating to anthropometric dimensions of the user

a. e. b. d
h. g. f. c
2 Pi-terms relating to geometric variables
H T . L D . W T.
 

H .L .W
l
a
w
a
 
:

L W
H B . H C . R v. 
H .L .W. 
 




H A. W b . R h 
h f . l f .wf 
3 Pi-term relating to environmental variable
6 = 
42
4Pi terms relating to dependent variables
Applying the Buckingham’s pi theorem we get,
7
=
vi.
(Dbt)j.
(Heat)k.
Wim .



EE
EE
V . Wi
8 = vi. (Dbt)j. (Heat)k. (Wi)m. HE
L0M0T0 = (L/T)i *  j * (ML2/T2)k* (ML/T2)m * (ML2/T2)
Equating the indices of L, M, T, and  on both sides of the above
equation we get,
 
HE
Heat
 
P . Heat
V . Wi
43
2 Test planning
This comprises of deciding test envelope, test points,
test sequence and experimental plan for the deduced set
of dimensional equations.
Determination of test envelope
i) Range of 1 , 2, 3, 5 are determined by putting
limiting values of numerator and denominator terms.
ii)ii) Range of 6 obtained from the meteorological data.
44
2 Determination of test points
3 Determination of test sequence
. In this test like majority of engineering
experiments partial randomized sequence
is proposed.
4 Determination of plan of experimentation
Design of Experimental set up
45
Experimental plan
Run
1
2
3
4
5
6
7
8
9
10
11
12
Level of ‘A’
A1
A1
A1
A1
A1
A1
A1
A1
A1
A1
A1
A1
Level of ‘G’
G1
G1
G1
G2
G2
G2
G3
G3
G3
G4
G4
G4
Level of ‘E’
E1morning
E1 noon
E1 afternoon
E1morning
E1 noon
E1 afternoon
E1morning
E1 noon
E1 afternoon
E1morning
E1 noon
E1 afternoon
46
47