1 / 15

CE 150 Fluid Mechanics

CE 150 Fluid Mechanics. G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico. Dimensional Analysis and Modeling. Reading: Munson, et al., Chapter 7. Introduction.

aline-battle
Download Presentation

CE 150 Fluid Mechanics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CE 150Fluid Mechanics G.A. Kallio Dept. of Mechanical Engineering, Mechatronic Engineering & Manufacturing Technology California State University, Chico CE 150

  2. Dimensional Analysis and Modeling Reading: Munson, et al., Chapter 7 CE 150

  3. Introduction • The solutions to most fluid mechanics problems involving real fluids require both analysis and experimental data • In this section, we look at the techniques used in designing experiments and correlating data • Specifically, we will learn how laboratory experiments (or models) can be used to describe similar phenomena outside the laboratory CE 150

  4. Dimensional Analysis Example • Consider an experiment that investigates the pressure drop in an incompressible Newtonian fluid flowing through a long, smooth-walled circular pipe • Based upon our “experience”, the pressure drop per unit length is CE 150

  5. Dimensional Analysis Example • To determine the nature of this function, an experiment could be designed which isolates and measures the effect of each variable: CE 150

  6. Buckingham Pi Theorem • Isolating the variables and performing this experiment would be difficult and time-consuming • The number of independent variables can be reduced by a dimensional analysis technique known as the Buckingham pi theorem: A dimensionally homogeneous equation with k variables can be reduced to k - r dimensionless products, where r is the minimum no. of reference dimensions needed to describe the variables CE 150

  7. Determining the Pi Terms • Method of Repeating Variables: 1) List all variables in the problem 2) Express each variable in terms of basic dimensions 3) Determine the number of pi terms 4) Select a number of repeating variables that equals the no. of reference dimensions 5) Form a pi term for each non-repeating variable such that the combination is dimensionless 6) Write an expression as a relationship of pi terms and consider its meaning CE 150

  8. Comments on Dimensional Analysis • Selection of variables • no simple procedure • requires understanding of the phenomena and physical laws • variables can be categorized by geometry, material properties, and external effects • Basic dimensions • usually use MLT or FLT • all three not always required • occasionally, the no. of reference dimensions is less than no. of basic dimensions required CE 150

  9. Comments on Dimensional Analysis • Repeating variables • number must equal the no. of reference dimensions • must include all basic dimensions contained in variables • must be dimensionally independent of each other • Pi terms • number of terms is unique but form of each term is not unique, since selection of repeating variables is somewhat arbitrary (unless there is only one pi term) CE 150

  10. Dimensionless Groups in Fluid Mechanics • Common groups given in Table 7.1 • Reynolds number • Froude number • Euler number • Mach number • Strouhal number • Weber number CE 150

  11. Correlation of Experimental Data • Dimensional analysis and experimental data can be used together to determine the specific relationship between pi terms • Problems with one pi term: • relationship is determined by dimensional analysis but constant must be determined by experiment CE 150

  12. Correlation of Experimental Data • Problems with two or more pi terms: • typically requires an experiment where one variable in each pi term is varied and its effect on the dependent variable is measured • an empirical mathematical relationship can be developed if the data shows good correlation • the mathematical relationship, or correlation equation, is only valid for the range of pi values tested CE 150

  13. Modeling & Similitude • A model is a representation of a physical system used to predict the behavior ofthe system • mathematical model • computer model • physical model, usually of different size and operating conditions • Similitude refers to ensuring that the results of the model study are similar to the actual physical system CE 150

  14. Theory of Models • For a given physical system (i.e., the prototype): • For a model of this system: • where the formof the function  will be the same as long as the physical phenomena are the same CE 150

  15. Theory of Models • If the model is designed and operated such that • Thus, if all pi terms are equal, then the measured value of 1mfor the model will be equal to the corresponding 1 for the prototype. These equations provide the modeling laws that will ensure similarity between prototype and model. CE 150

More Related