ES 202 Fluid and Thermal Systems Lab 1: Dimensional Analysis

1. ES 202Fluid and Thermal SystemsLab 1:Dimensional Analysis

2. 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

3. 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.

4. 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 • Dueby 5 pm one week after the lab at my office (O-219)

5. 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)

6. 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

7. Dimensional Homogeneity • The dimension on both sides of any physically meaningful equation must be the same. • Do exercises on Page 2 of Lab 1

8. 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.

9. 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.

10. 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 (P2, P3, P4, …………..., 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

11. What is the Procedure? • Come up with the list of dependent and independent variables (the least trivial part in my opinion) • Identify the number of reference dimensions represented by this set of variables which gives the value of r • Choose a set of rrepeating variables (these rrepeating variables should span all the reference dimensions in the problem) • All the remaining k - r variables are automatically the non-repeating variables

12. Continuation of Procedure • 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 • By invoking dimensional homogeneity on both sides of the equation, the values of the unknown exponents can be found • Repeat the P group formulation for each of the non-repeating variables

13. 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

14. 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?

15. Any Advantages?? • Absolutely “YES” • You may reduce a thick pile of graphs to a singlexy-plot • For examples: • 4 variables in 3 dimensions can be reduced to 1P group which is equal to a constant (dimensionless) • 5 variables in 3 dimensions can be reduced to 2P groups taking the general form P 1 = f (P 2 )

16. Drag Coefficient for a Sphere taken from Figure 8.2 in Fluid Mechanics by Kundu

17. More Exercises • Sliding block • Pendulum

18. 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!

19. 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.

20. Complete Similarity • Model versus prototype (full scale) • Geometric similarity • Kinematic similarity • Dynamic similarity

21. Central Theme The Dimensionless world is simpler!!

23. More Examples • Sliding block • Pendulum • Nuclear bomb • Terminal velocity of a falling object • Pressure drop along a pipe