ES 202Fluid and Thermal Systems

Lab 1:Dimensional Analysis

Road Map of Lab 1

• Announcements

• Guidelines on write-up

• Fundamental of dimensional analysis – 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.rose-hulman.edu/Class/me/ES202

• Due by 5 pm one week after the lab at my office (O-219)

Dimension Versus Unit

• Dimensions (units)– 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 ( 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

1 = g ( 2, 3, 4, …………..., k-r )

where 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 group by forming product of one of the non-repeating variables and all the repeating variables raised to some unknown powers. For example,

= y x1a x2

b x3c

6) By invoking dimensional homogeneity on both sides of the equation, the values of the unknown exponents can be found

7) Repeat the group formulation for each of the non-repeating variables

Properties of Groups

• The groups are not unique (depend on your choice of repeating variables)

• Any combinations of groups can generate another group

• The simpler 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 group which is equal to a constant (dimensionless)

– 5 variables in 3 dimensions can be reduced to 2 groups taking the general form

1 = f (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 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 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