Transcript ES 202 Fluid and Thermal Systems Lab 1: Dimensional Analysis

ES 202
Fluid and Thermal Systems
Lab 1:
Dimensional Analysis
Road Map of Lab 1
• Announcements
• Guidelines on write-up
• Fundamental of dimensional analysis
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difference between “dimension” and “unit”
primary (fundamental) versus secondary (derived)
functional dependency of data
Buckingham Pi Theorem
alternative way of data representation (reduction)
active learning exercises
Announcements
• Lab 2 will be at Olin 110 (4th week)
• Lab 3 will be at DL 205 (8th week)
• You are not required to hand in the in-class lab
exercises.
About the Write-Up
• Raw data sheet and write-up format for Lab 1
can be downloaded at
http://www.rosehulman.edu/Class/me/ES202
• Due by 5 pm one week after the lab at my
office (O-219)
Dimension Versus Unit
• Dimensions (units)
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Length (m, ft)
Mass (kg, lbm) MLT system
Time (sec, minute, hour)
Force (N, lbf) FLT system
Temperature (deg C, deg F, K, R)
Current (Ampere)
Primary Versus Secondary
• In the MLT system, the dimension of Force is derived
from Newton’s law of motion.
• In the FLT system, the dimension of Mass is derived
likewise.
• Quantities like Pressure and Charge can be derived
based on their respective definitions.
• Do exercises on Page 1 of Lab 1
Dimensional Homogeneity
• The dimension on both sides of any physically
meaningful equation must be the same.
• Do exercises on Page 2 of Lab 1
Data Representation
• Given a functional dependency
y = f (x1, x2, x3, …………..., xk )
where y is the dependent variable while all the xi’s are
the independent ones. Both y and the xi’s can be
dimensional or dimensionless.
• One way to express the functional dependency is to
view the above relation as an n-dimensional problem:
to plot the dependency of y against any one of the xi’s
while keeping the remaining ones fixed.
Buckingham Pi Theorem
If an equation involving k variables is
dimensionally homogeneous, it can be
reduced to a relationship among k - r
independent products (P groups),
where r is the minimum number of
reference dimensions required to
describe the variables.
Alternative Way of Data Representation
• It is advantageous to view the same functional
dependency in a smaller dimensional space
• Cast
y = f (x1, x2, x3, …………..., xk-1 )
into
P 1 = g (P 2, P 3, P 4, …………..., P k-r )
where P i’s are non-dimensional groups formed by
combining y and the xi’s, and r is the number of
reference dimensions building the xi’s
What is the Procedure?
1) Come up with the list of dependent and independent
variables (the least trivial part in my opinion)
2) Identify the number of reference dimensions
represented by this set of variables which gives the
value of r
3) Choose a set of r repeating variables (these r
repeating variables should span all the reference
dimensions in the problem)
4) All the remaining k - r variables are automatically
the non-repeating variables
Continuation of Procedure
5) Form each P group by forming product of one of
the non-repeating variables and all the repeating
variables raised to some unknown powers. For
example,
P = y x1a x2b x3c
6) By invoking dimensional homogeneity on both
sides of the equation, the values of the unknown
exponents can be found
7) Repeat the P group formulation for each of the
non-repeating variables
Properties of P Groups
• The P groups are not unique (depend on your
choice of repeating variables)
• Any combinations of P groups can generate
another P group
• The simpler P groups are the preferred choices
Motivational Exercise
• Drag on a tennis ball
– work out the whole problem
– what if it is not spherical, say oval?
– what if it is not placed parallel to flow direction
but at an angle?
Any Advantages??
• Absolutely “YES”
• You may reduce a thick pile of graphs to a
single xy-plot
• For examples:
– 4 variables in 3 dimensions can be reduced to 1 P
group which is equal to a constant (dimensionless)
– 5 variables in 3 dimensions can be reduced to 2 P
groups taking the general form
P 1 = f (P 2 )
Drag Coefficient for a Sphere
taken from Figure 8.2 in Fluid Mechanics by Kundu
More Exercises
• Sliding block
• Pendulum
What is the Key Point?
• There are more than one way to view the same
physical problem.
• Some ways are more economical than others
• The reduction of dimensions from the physical
dimensional variables to non-dimensional P
groups is significant!
Reflection on the Procedures
• The most important step is to come up with the list of
independent variables (Buckingham cannot help in
this step!)
• Once the dependent and independent variables are
determined (based on a combination of judgment,
intuition and experience), the rest is just routine, i.e.
finding the P groups!
• However, Buckingham cannot give you the exact
form of the functional dependency. It has to come
from experiments, models or simulations.
Complete Similarity
• Model versus prototype (full scale)
• Geometric similarity
• Kinematic similarity
• Dynamic similarity
Central Theme
The Dimensionless world is simpler!!
More Examples
• Sliding block
• Pendulum
• Nuclear bomb
• Terminal velocity of a falling object
• Pressure drop along a pipe