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

12/12/2006BIEN 301 – Winter Pressure  Definition (White 2.1) A force applied over a surface area. 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. In fluid statics, the forces exerted by the pressures on each of the faces of the fluid element must sum to zero.

12/12/2006BIEN 301 – Winter Pressure From this formula, we can see that a few principal facts fall out: From this formula, we can see that a few principal facts fall out: Hydrostatic pressure is a function of height of the fluid columnHydrostatic pressure is a function of height of the fluid column The pressure in any plane normal to the gravitational field is identical.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.For hydrostatic conditions, the forces generated by the fluid must come from the gravitationally-induced weight component only.

12/12/2006BIEN 301 – Winter Pressure What about if the fluid is moving? What about if the fluid is moving? Strain rates will exist, and they will be out of balance.Strain rates will exist, and they will be out of balance. Viscous stresses will existViscous stresses will exist

12/12/2006BIEN 301 – Winter Pressure  Forces from Pressure From our definition, pressure is a force over a surface area. From our definition, pressure is a force over a surface area. What happens when we get pressure variations? What happens when we get pressure variations? What causes 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. Because pressure variations can come from many different sources, we need a more generalized expression for expressing these forces.

12/12/2006BIEN 301 – Winter Pressure  Surface and Body Forces Two types of forces that can act on our fluid element Two types of forces that can act on our fluid element Can you think of examples of surface vs. body forces? Can you think of examples of surface vs. body forces? Principal body force we’ll deal with here: Principal body force we’ll deal with here: GRAVITYGRAVITY Integrating the force from gravitational effects over our element volume:Integrating the force from gravitational effects over our element volume:

12/12/2006BIEN 301 – Winter Pressure  We’re missing one principal force Surface forces due to viscous effects Surface forces due to viscous effects

12/12/2006BIEN 301 – Winter 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.

12/12/2006BIEN 301 – Winter 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:

12/12/2006BIEN 301 – Winter 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

12/12/2006BIEN 301 – Winter Pressure  Flow under balanced forces (ΣF=0) Hydrostatic Hydrostatic Steady motion Steady motion

12/12/2006BIEN 301 – Winter Pressure  Rigid Body Motion, drops out the viscous term Rotation Rotation Translation Translation Examples? Why does the viscous term disappear? Examples? Why does the viscous term disappear?

12/12/2006BIEN 301 – Winter Pressure  Irrotational Motion Different than rigid body flow…why? Different than rigid body flow…why? Does not have to behave like a rigid body Does not have to behave like a rigid body There simply are no rotational terms. There simply are no rotational terms.

12/12/2006BIEN 301 – Winter Pressure Measurement  Absolute, Vacuum, and Gauge pressure Can anyone tell me what the most commonly used pressure reporting method is? Can anyone tell me what the most commonly used pressure reporting method is? Is there a better scientific way of reporting the pressure? Is there a better scientific way of reporting the pressure? Why? Why?

12/12/2006BIEN 301 – Winter Pressure Measurement  Hydrostatic condition As we mentioned before, the hydrostatic (no motion, or balanced forces) condition reduces to: As we mentioned before, the hydrostatic (no motion, or balanced forces) condition reduces to:  For a fluid at rest, the horizontal components drop out.

12/12/2006BIEN 301 – Winter Pressure Measurement  We can also express this in terms of an integral

12/12/2006BIEN 301 – Winter Pressure Measurement  For liquids, incompressibility is a good assumption

12/12/2006BIEN 301 – Winter 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)

12/12/2006BIEN 301 – Winter Pressure Measurement  What if the medium is a gas (compressible)?  How about introducing the ideal gas law?  A sufficiently accurate assumption for most cases.

12/12/2006BIEN 301 – Winter 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?

12/12/2006BIEN 301 – Winter Pressure Measurement  Common pressure measurement instrument designations: Gravity-based: Gravity-based: barometer, manometer, dead weight pistonbarometer, manometer, dead weight piston Elastic deformation: Elastic deformation: bourdon tube, diaphragm, bellows, strain gauge, displaced optical beamsbourdon tube, diaphragm, bellows, strain gauge, displaced optical beams Gas behavior: Gas behavior: Gas compression, thermal conductance, molecular impact, ionization, thermal conductivity.Gas compression, thermal conductance, molecular impact, ionization, thermal conductivity. Electric output: Electric output: Resistance, diffused strain gauge, piezoelectric, potentiometric, magnetic inductance, linear variable differential transformer, resonant frequency.Resistance, diffused strain gauge, piezoelectric, potentiometric, magnetic inductance, linear variable differential transformer, resonant frequency. Luminescent coatings Luminescent coatings Surface PressuresSurface Pressures

12/12/2006BIEN 301 – Winter Pressure Measurement  Common pressure measurement instrument designations: Electric output: Electric output: Resistance, diffused strain gauge, piezoelectric, potentiometric, magnetic inductance, linear variable differential transformer, resonant frequency.Resistance, diffused strain gauge, piezoelectric, potentiometric, magnetic inductance, linear variable differential transformer, resonant frequency. Luminescent coatings Luminescent coatings Surface PressuresSurface Pressures  Examples of where we might choose different pressure measurement devices in biomedical applications?

12/12/2006BIEN 301 – Winter Pressure Measurement  Let’s apply what we’ve learned to some flow examples: Ex. 2.4 Ex. 2.4 Ex. 2.7 Ex. 2.7 P2.23 P2.23

12/12/2006BIEN 301 – Winter Assignment  HW 4 has been posted on blackboard  Exam 1 Reviews have been posted Reviews have been posted In-class exam materials allowed: In-class exam materials allowed: 1 Calculator1 Calculator Writing ImplementsWriting Implements Chapter Reviews from course documents, NO NOTES.Chapter Reviews from course documents, NO NOTES. Lecture slides, 6 to a page, NO NOTES.Lecture slides, 6 to a page, NO NOTES. Fluid Mechanics, fifth edition, by White (class textbook), 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.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 Take-Home Option Sign-Up due by Wednesday, 11:59 pm.Sign-Up due by Wednesday, 11:59 pm.

12/12/2006BIEN 301 – Winter Questions?