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Fluid Mechanics and Energy Transport BIEN 301 Lecture 2 Introduction to Fluids, Flow Fields, and Dimensional Analysis Juan M. Lopez, E.I.T. Research Consultant LeTourneau University Adjunct Lecturer Louisiana Tech University

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12/05/2006BIEN 301 – Winter 2006-2007 History of Fluid Mechanics White 1.14 shows us how Fluid Mechanics has evolved in a helical fashion, returning to its roots, with improvements each time. Pre-historic and early history aqueducts and waterworks – Empirically Designed and Built Pre-historic and early history aqueducts and waterworks – Empirically Designed and Built Archimedes (200’s B.C.) and Buoyancy / Vector addition – Theoretical work with Experimental roots Archimedes (200’s B.C.) and Buoyancy / Vector addition – Theoretical work with Experimental roots 200’s B.C. to Renaissance ship and canal building – Empirical advances, no great amount of experimental work 200’s B.C. to Renaissance ship and canal building – Empirical advances, no great amount of experimental work Leonardo da Vinci first formulated the one-dimensional conservation of mass equation – Theoretical stemming from empirical observations. Leonardo da Vinci first formulated the one-dimensional conservation of mass equation – Theoretical stemming from empirical observations.

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12/05/2006BIEN 301 – Winter 2006-2007 History of Fluid Mechanics Mariotte (1600’s) built the first wind tunnel – Testing theoretical ideas with experimental work. Mariotte (1600’s) built the first wind tunnel – Testing theoretical ideas with experimental work. Isaac Newton (1600’s-1700’s) generated the mathematics which allowed fluid momentum to be studied. Isaac Newton (1600’s-1700’s) generated the mathematics which allowed fluid momentum to be studied. Bernoulli, D’Alembert, Euler, Lagrange, Laplace, all developed their work in frictionless fluids, and showed the need for a formulation that would do away with the paradox of an object with no drag immersed in a moving stream, a natural result of frictionless fluid assumptions – Theoretical advances mostly. Bernoulli, D’Alembert, Euler, Lagrange, Laplace, all developed their work in frictionless fluids, and showed the need for a formulation that would do away with the paradox of an object with no drag immersed in a moving stream, a natural result of frictionless fluid assumptions – Theoretical advances mostly. These theoretical results were unsatisfactory to engineers, so as a natural backlash, hydraulics was developed as an almost purely experimental form by Pitot, Borda, Poiseuille, etc. These theoretical results were unsatisfactory to engineers, so as a natural backlash, hydraulics was developed as an almost purely experimental form by Pitot, Borda, Poiseuille, etc.

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12/05/2006BIEN 301 – Winter 2006-2007 History of Fluid Mechanics Late 1800’s, finally there was a trend towards the unification between experimental hydraulics and theoretical hydrodynamics by the likes of Froude, Raylegh, and Reynolds. All of these gentlemen have dimensionless groups named after them due to the importance of their work. Late 1800’s, finally there was a trend towards the unification between experimental hydraulics and theoretical hydrodynamics by the likes of Froude, Raylegh, and Reynolds. All of these gentlemen have dimensionless groups named after them due to the importance of their work. Navier and Stokes began to more fully explore viscous flow in the mid to late 1800’s, setting the stage for Prandtl. Navier and Stokes began to more fully explore viscous flow in the mid to late 1800’s, setting the stage for Prandtl. In the early 1900’s, Prandtl developed boundary layer theory, one of the most important advances in fluid mechanics, identified by White as the single most important tool in modern flow analysis. In the early 1900’s, Prandtl developed boundary layer theory, one of the most important advances in fluid mechanics, identified by White as the single most important tool in modern flow analysis.

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12/05/2006BIEN 301 – Winter 2006-2007 History of Fluid Mechanics The past tied to the present These past examples of development in fluid mechanics remain important due to the individual contributions each advance has made to our current understanding. These past examples of development in fluid mechanics remain important due to the individual contributions each advance has made to our current understanding. In fact, we continue to study many of these individual ideas as simplified examples of fluid behavior. In fact, we continue to study many of these individual ideas as simplified examples of fluid behavior. Fluid mechanics encompasses almost every field of physical systems, and a basic understanding of the mathematics, terminology, and usage will greatly benefit you in any engineering field. Fluid mechanics encompasses almost every field of physical systems, and a basic understanding of the mathematics, terminology, and usage will greatly benefit you in any engineering field.

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12/05/2006BIEN 301 – Winter 2006-2007 What is a Fluid? Matter that is unable to resist shear by a static deflection. (White, 1.2) Fluid will deflect under shear unless opposed by some external force. The rate of strain to stress is dependent on the viscosity of the fluid.

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12/05/2006BIEN 301 – Winter 2006-2007 What is a Fluid? This lack of resistance to shear explains why fluid take the shape of their containers, or spill when there is no body to contain them.

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12/05/2006BIEN 301 – Winter 2006-2007 What is a Fluid? Mechanical Description – Mohr’s Circle

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12/05/2006BIEN 301 – Winter 2006-2007 What is a Fluid? As with everything, we make some assumptions in our definition- Continuum (White 1.3) Continuum (White 1.3) Infinitely Divisible – All divisions have same properties in homogeneous fluidInfinitely Divisible – All divisions have same properties in homogeneous fluid For real systems, there are uncertainties brought about by volumes that are too small or too large.For real systems, there are uncertainties brought about by volumes that are too small or too large. Physical properties are defined and have finite values throughout the continuum Physical properties are defined and have finite values throughout the continuum Thermal properties are defined and have finite values throughout the continuum Thermal properties are defined and have finite values throughout the continuum

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12/05/2006BIEN 301 – Winter 2006-2007 Dimensions vs. Units We must inherently have a way to describe the systems we are studying. We describe these systems with Dimensions and quantify these dimensions with Units. Four primary Dimensions in our study of Fluid Mechanics: Mass, {M} Mass, {M} Length, {L} Length, {L} Time, {T} Time, {T} Temperature, {Θ} Temperature, {Θ}

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12/05/2006BIEN 301 – Winter 2006-2007 Dimensions vs. Units It is imperative that you learn consistency in your dimensional analysis. Fluid mechanics lends itself to some extremely awkward units, especially in the British system. For this course, we will primarily stick with the International System (SI), but we will refresh our memories from time to time on how to interact with the British Gravitational (BG) units. The use of tables is an inherent task in engineering work. Become familiar with the tables such as White, Table 1.1, 1.2, and Appendix Tables A.1-A.6, and how to properly use them.

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12/05/2006BIEN 301 – Winter 2006-2007 Dimensions vs. Units Using the tables, perform the following conversion:

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12/05/2006BIEN 301 – Winter 2006-2007 Dimensions vs. Units

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12/05/2006BIEN 301 – Winter 2006-2007 Dimensional Consistency Dimensional Homogeneity (White 1.4) Theoretical Equations – dimensionally homogeneous Theoretical Equations – dimensionally homogeneous

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12/05/2006BIEN 301 – Winter 2006-2007 Dimensional Consistency However, much work in fluid mechanics has been empirical, and this can lead to problematic situations.

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12/05/2006BIEN 301 – Winter 2006-2007 Uncertainty Once we have established a way to describe these systems, we must also account for the uncertainty in our experimentation. (White 1.11) Instruments and all physical measurements have some form of uncertainty. Accounting for all the measurements is important Accounting for all the measurements is important Adding them all is simply not realistic Adding them all is simply not realistic A simplified Root Mean Square (RMS) approach is recommended. A simplified Root Mean Square (RMS) approach is recommended.

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12/05/2006BIEN 301 – Winter 2006-2007 Uncertainty RMS Formulation:

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12/05/2006BIEN 301 – Winter 2006-2007 Uncertainty RMS Example

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12/05/2006BIEN 301 – Winter 2006-2007 Basic Physical Properties Thermodynamics (White 1.6) Principal components of velocity vectors Principal components of velocity vectors Pressure, pPressure, p Density, ρDensity, ρ Temperature, TTemperature, T Principal components of work, heat, and energy balance. Principal components of work, heat, and energy balance. Internal Energy, ûInternal Energy, û Enthalpy, h = û + ρ/pEnthalpy, h = û + ρ/p Principal transport properties Principal transport properties Viscosity, μViscosity, μ Thermal Conductivity, kThermal Conductivity, k Together, these define the state of the fluid.

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12/05/2006BIEN 301 – Winter 2006-2007 Basic Physical Properties Additional Properties (White 1.6) Specific Weight, γ = ρg Specific Weight, γ = ρg Specific Gravity Specific Gravity SG gas = ρ gas / ρ airSG gas = ρ gas / ρ air SG water = ρ liquid / ρ waterSG water = ρ liquid / ρ water Potential Energy Potential Energy -g●r-g●r Kinetic Energy Kinetic Energy 0.5 V 20.5 V 2 Total Energy Total Energy e = û + 0.5 V 2 + (-g●r)e = û + 0.5 V 2 + (-g●r)

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12/05/2006BIEN 301 – Winter 2006-2007 State Relationships State Relationships for Gases (White 1.6) Thermodynamic properties are related to each other by state relationships. For gases, there is the ideal gas law (perfect-gas law). Thermodynamic properties are related to each other by state relationships. For gases, there is the ideal gas law (perfect-gas law). p = ρRT where R = c p – c v (gas constant)p = ρRT where R = c p – c v (gas constant) The gas constant is related to the universal gas constant, Λ by the following equation: The gas constant is related to the universal gas constant, Λ by the following equation: Λ = R gas * M gasΛ = R gas * M gas

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12/05/2006BIEN 301 – Winter 2006-2007 State Relationships State Relationships for Liquids No direct analog of the ideal gas law exists for liquids. No direct analog of the ideal gas law exists for liquids. Why? If fluids involves liquids and gases, why can we not get a direct correlation to a liquid form? Why? If fluids involves liquids and gases, why can we not get a direct correlation to a liquid form? Compressibility. The ideal gas law assumes compressibility, whereas most liquids are mostly incompressible.Compressibility. The ideal gas law assumes compressibility, whereas most liquids are mostly incompressible.

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12/05/2006BIEN 301 – Winter 2006-2007 State Relationships State Relationships for Liquids As an example of this lack of direct relationship, see from White, eq. 1.19: As an example of this lack of direct relationship, see from White, eq. 1.19: Where B and n are dimensionless parameters that vary with temperature. Where B and n are dimensionless parameters that vary with temperature.

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12/05/2006BIEN 301 – Winter 2006-2007 Velocity Fields For many of the problems encountered here, the velocity field will be the solution to our given problem, or an integral part thereof. (White 1.5) The three-dimensional velocity field can be expressed in a variety of ways:

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12/05/2006BIEN 301 – Winter 2006-2007 Velocity Fields Simplified problems: in White, example 1.5, we see the convective result for a 1-Dimensional problem. The extended answer for the 3D problem is as follows:

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12/05/2006BIEN 301 – Winter 2006-2007 Velocity Fields Dealing with partial differential equations. Cross out terms ahead of time, simplifies calculations. Cross out terms ahead of time, simplifies calculations. For the 2D problem, there are no velocity components in the Z direction (no w magnitude, and no δ() /δz. For the 2D problem, there are no velocity components in the Z direction (no w magnitude, and no δ() /δz.

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12/05/2006BIEN 301 – Winter 2006-2007 Velocity Fields

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12/05/2006BIEN 301 – Winter 2006-2007 Application So, what can we do with all of this stuff? Why re- hash over so many of the basics we have seen in other courses over the years? While we may have been exposed to all of these concepts, they become integral in the study of fluid mechanics. While we may have been exposed to all of these concepts, they become integral in the study of fluid mechanics. Familiarity with these ideas is no longer enough, we must master these concepts and learn to apply them in new and effective ways. Familiarity with these ideas is no longer enough, we must master these concepts and learn to apply them in new and effective ways.

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12/05/2006BIEN 301 – Winter 2006-2007 Application With these basics we will be able to: Fully describe and define the subject of our study: Fluids. Fully describe and define the subject of our study: Fluids. Perform dimensionally consistent calculations, increasing the skill set required of a modern professional engineer. Perform dimensionally consistent calculations, increasing the skill set required of a modern professional engineer. Be conversant and capable in both the BG and the SI system, able to convert between the two as the problem requires. Be conversant and capable in both the BG and the SI system, able to convert between the two as the problem requires.

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12/05/2006BIEN 301 – Winter 2006-2007 Application With these basics we will be able to: Understand the basic thermodynamic concepts required to extend our analysis from pure fluid mechanics to true energy transport problems. Understand the basic thermodynamic concepts required to extend our analysis from pure fluid mechanics to true energy transport problems. Heat TransferHeat Transfer Temperature-dependent effectsTemperature-dependent effects Accurately and professionally report our findings, accounting for our experimental error and/or uncertainty. Accurately and professionally report our findings, accounting for our experimental error and/or uncertainty.

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12/05/2006BIEN 301 – Winter 2006-2007 Application As was mentioned before: these skills, though ideally common throughout all of our engineering courses, become absolute cornerstones of success for a subject as complex and difficult as fluid mechanics.

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12/05/2006BIEN 301 – Winter 2006-2007 Fundamental Approaches There are two primary approaches to problem solving in fluid mechanics: Lagrangian and Eulerian Lagrangian and Eulerian Lagrangian: follows a fluid particle as it moves through a flow field.Lagrangian: follows a fluid particle as it moves through a flow field. Eulerian: Observes passing fluid particles from a stationary position relative to the flow fieldEulerian: Observes passing fluid particles from a stationary position relative to the flow field

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12/05/2006BIEN 301 – Winter 2006-2007 Fundamental Approaches Examples: Lagrangian – Lagrangian – A user observes traffic on the freeway as he sits in his vehicle, travelling down the freeway along with the traffic. Traffic jams, velocity changes, etc, are all marked and observed to attempt to describe the flow of traffic through a section of freeway.A user observes traffic on the freeway as he sits in his vehicle, travelling down the freeway along with the traffic. Traffic jams, velocity changes, etc, are all marked and observed to attempt to describe the flow of traffic through a section of freeway. Eulerian – Eulerian – A state trooper monitors freeway traffic from a hidden location under the bridge, monitoring for changes in traffic that could indicate potential trouble. Multiple state troopers and cameras along the road give a “big picture” perspective to traffic managers.A state trooper monitors freeway traffic from a hidden location under the bridge, monitoring for changes in traffic that could indicate potential trouble. Multiple state troopers and cameras along the road give a “big picture” perspective to traffic managers.

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12/05/2006BIEN 301 – Winter 2006-2007 Fundamental Approaches Eulerian will be our fundamental approach for this course. Probes at different points in the fluid stream are much more easy to design and monitor for smaller systems that we’ll concern ourselves with than large instrumentation designs that follow the flow. Can you think of an example of Eulerian monitoring and/or Lagrangian monitoring in biomedical systems? What are some potential benefits of each type of system relative to this application?

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12/05/2006BIEN 301 – Winter 2006-2007 Assignment HW 2 has been posted on blackboard Project Proposals due soon! Individual project sign-ups will be available by tonight on blackboard.

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12/05/2006BIEN 301 – Winter 2006-2007 Questions?

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