Juan M. Lopez, E.I.T. Research Consultant LeTourneau University Adjunct Lecturer

1. Fluid Mechanics and Energy TransportBIEN 301Lecture 4 Pressure Distribution, Hydrostatic Forces, and Pressure Measurement Juan M. Lopez, E.I.T. Research Consultant LeTourneau University Adjunct Lecturer Louisiana Tech University

2. Pressure • Definition (White 2.1) • A force applied over a surface area. • In fluid statics, the forces exerted by the pressures on each of the faces of the fluid element must sum to zero. BIEN 301 – Winter 2006-2007

3. Pressure • From this formula, we can see that a few principal facts fall out: • Hydrostatic pressure is a function of height of the fluid column • The pressure in any plane normal to the gravitational field is identical. • For hydrostatic conditions, the forces generated by the fluid must come from the gravitationally-induced weight component only. BIEN 301 – Winter 2006-2007

4. Pressure • What about if the fluid is moving? • Strain rates will exist, and they will be out of balance. • Viscous stresses will exist BIEN 301 – Winter 2006-2007

5. Pressure • Forces from Pressure • From our definition, pressure is a force over a surface area. • What happens when we get pressure variations? • What causes pressure variations? • Because pressure variations can come from many different sources, we need a more generalized expression for expressing these forces. BIEN 301 – Winter 2006-2007

6. Pressure • Surface and Body Forces • Two types of forces that can act on our fluid element • Can you think of examples of surface vs. body forces? • Principal body force we’ll deal with here: • GRAVITY • Integrating the force from gravitational effects over our element volume: BIEN 301 – Winter 2006-2007

7. Pressure • We’re missing one principal force • Surface forces due to viscous effects BIEN 301 – Winter 2006-2007

8. Pressure • We can now combine all of our defined terms to generate a more general expression for the balance of forces on a fluid element. • This is a form of the differential momentum equation from Chapter 4. BIEN 301 – Winter 2006-2007

9. Pressure • For now, we assume that we know the velocity and acceleration acting on our fluid. If we have an acceleration vector, a, we can then re-express our equation as follows: BIEN 301 – Winter 2006-2007

10. Pressure • With the knowns (V, a), we can solve for the pressure field via direct integration. • This form will be very useful. To illustrate this, we’ll take a few examples BIEN 301 – Winter 2006-2007

11. Pressure • Flow under balanced forces (ΣF=0) • Hydrostatic • Steady motion BIEN 301 – Winter 2006-2007

12. Pressure • Rigid Body Motion, drops out the viscous term • Rotation • Translation • Examples? Why does the viscous term disappear? BIEN 301 – Winter 2006-2007

13. Pressure • Irrotational Motion • Different than rigid body flow…why? • Does not have to behave like a rigid body • There simply are no rotational terms. BIEN 301 – Winter 2006-2007

14. Pressure Measurement • Absolute, Vacuum, and Gauge pressure • Can anyone tell me what the most commonly used pressure reporting method is? • Is there a better scientific way of reporting the pressure? • Why? BIEN 301 – Winter 2006-2007

15. Pressure Measurement • Hydrostatic condition • As we mentioned before, the hydrostatic (no motion, or balanced forces) condition reduces to: • For a fluid at rest, the horizontal components drop out. BIEN 301 – Winter 2006-2007

16. Pressure Measurement • We can also express this in terms of an integral BIEN 301 – Winter 2006-2007

17. Pressure Measurement • For liquids, incompressibility is a good assumption BIEN 301 – Winter 2006-2007

18. Pressure Measurement • It is this simple relationship which is exploited in many pressure measurement devices and/or calculations. • For example, calculating the rise of mercury in a tube (a mercury barometer) BIEN 301 – Winter 2006-2007

19. Pressure Measurement • What if the medium is a gas (compressible)? • How about introducing the ideal gas law? • A sufficiently accurate assumption for most cases. BIEN 301 – Winter 2006-2007

20. Pressure Measurement • Pressure is not something we can measure directly, but rather derive it from some other measurement. • What forms of measurement of pressure are used, and how do we have to derive the signal from these? BIEN 301 – Winter 2006-2007

21. Pressure Measurement • Common pressure measurement instrument designations: • Gravity-based: • barometer, manometer, dead weight piston • Elastic deformation: • bourdon tube, diaphragm, bellows, strain gauge, displaced optical beams • Gas behavior: • Gas compression, thermal conductance, molecular impact, ionization, thermal conductivity. • Electric output: • Resistance, diffused strain gauge, piezoelectric, potentiometric, magnetic inductance, linear variable differential transformer, resonant frequency. • Luminescent coatings • Surface Pressures BIEN 301 – Winter 2006-2007

22. Pressure Measurement • Common pressure measurement instrument designations: • Electric output: • Resistance, diffused strain gauge, piezoelectric, potentiometric, magnetic inductance, linear variable differential transformer, resonant frequency. • Luminescent coatings • Surface Pressures • Examples of where we might choose different pressure measurement devices in biomedical applications? BIEN 301 – Winter 2006-2007

23. Pressure Measurement • Let’s apply what we’ve learned to some flow examples: • Ex. 2.4 • Ex. 2.7 • P2.23 BIEN 301 – Winter 2006-2007

24. Assignment • HW 4 has been posted on blackboard • Exam 1 • Reviews have been posted • In-class exam materials allowed: • 1 Calculator • Writing Implements • Chapter Reviews from course documents, NO NOTES. • Lecture slides, 6 to a page, NO NOTES. • Fluid Mechanics, fifth edition, by White (class textbook), NO NOTES. • If I find handwritten notes in these sections, your materials will be removed. Any additional cheating will result in failing the test, and maybe the course. • Take-Home Option • Sign-Up E-mail due by Wednesday, 11:59 pm. BIEN 301 – Winter 2006-2007

25. Questions? BIEN 301 – Winter 2006-2007