3.8 The First Law of Thermodynamics and the Energy Equation

Size: px
Start display at page:

Download "3.8 The First Law of Thermodynamics and the Energy Equation"

Transcription

1 CEE 3310 Control Volume Analysis, Sep 30, Review Conservation of angular momentum 1-D form ( r F )ext = [ˆ ] ( r v)d + ( r v) out ṁ out ( r v) in ṁ in t CV 3.8 The First Law of Thermodynamics and the Energy Equation The first law of thermodynamics tells us: Time rate of change of total system energy = Time rate of change by heat transfer + Time rate of change by work transfer or ( ) de = dt Q Ẇ sys where E is energy, Q is heat, and W is work. Recall that ( ) indicates time rate of change. Sign conventions: Q is the transfer by radiation, conduction, convection of heat. Transfer into the control volume is positive. W > 0 is work done by the system on the surroundings and W < 0 is work done on the system by the surroundings. Note that it is not uncommon to use the opposite sign convention for work, in which case the equation is written with the final terms as +Ẇ

2 66 (you may have seen it this way in a thermodynamics course). In CEE 3310 we will always define W>0 as work done by the system on the surroundings. We can decompose the rate of work into components: Ẇ = Ẇshaft + Ẇpressure + Ẇviscous stress where W shaft is the work done by a machine such as a pump, turbine, piston, generator, etc. Ẇ shaft = T shaft Ω, where T shaft is the torque and Ω is the angular velocity. Note that gravitational work will enter our energy budget through potential energy, which will show up linearly related to the height above a datum Stress Induced Work per Time (ower) Work occurs by applying a force over a distance. Therefore pressure (normal stress) and shear (viscous stress) can produce work. If we look at the rate of work, or work per unit time, we have power and we see that we can think of this as applying a force on a system with a given velocity. δẇ = δ F v Therefore for pressure we have δẇpres = n δa v = v n δa Therefore ˆ Ẇ pres = v n da CS This term is usually moved to the right-hand-side flux term of the energy equation as it is a flux, which is how we will treat it.

3 CEE 3310 Control Volume Analysis, Sep 30, For shear stress we have δẇvisc = τδa v = τ v δa Therefore ˆ Ẇ visc = τ v da CS But shear is internally self-canceling. On the control surface v =0at solid boundaries (no-slip boundary condition), if the control surface is normal to the flow then τ v and hence τ v =0. Hence it is often reasonable to assume that the shear stress induced work is small. What is a good environmental example of when this assumption breaks down? Let s consider the amount of work done! 3.9 The Energy Equation With our definitions of work and energy we are now ready to apply the Reynolds Transport Theorem to produce the conservation of energy equation. Let B = E and β = e = E/m, the energy per unit mass. We can decompose e into the following components e =û + v2 2 + gz + other where û is the internal energy per unit mass, v 2 /2 is the kinetic energy per unit mass, and gz is the potential energy per unit mass. We will ignore other sources of internal energy but from the definition they are easily included.

4 68 We can write the Reynolds Transport Theorem flux term (including the pressure work term) ˆ CS (û + v2 2 + gz + ) ( v n) da Therefore ( ) de = dt Q Ẇshaft = d ˆ sys dt CV ) ˆ (û + v2 2 + gz d + (û + v2 CS 2 + gz + ) ( v n) da The 1-D form is Q Ẇshaft = d dt CV ( û + v2 2 + gz ) d + outflows ( ) û + v2 + gz + ṁ ( ) û + v2 + gz + ṁ 2 2 inflows We sometimes will choose to write (û + /) =ĥ = enthalpy Forms of the Energy Equation We often find it convenient to cast the energy equation in alternate forms. Velocity Squared form: If the flow is at steady state then conservation of mass gives us that ṁ out = ṁ in = ṁ and we can normalized all of our quantities by ṁ which leaves our homogeneous equation with terms having units of velocity squared Q ṁ Ẇshaft ṁ =û out + v2 out 2 + gz out + out û in v2 in 2 gz in in Defining q = we can write the above as Q heat transfer = ṁ unit mass and w s = Ẇshaft ṁ shaft work = unit mass û out + v2 out 2 + gz out + out =û in + v2 in 2 + gz in + in + q w s

5 CEE 3310 Control Volume Analysis, Sep 30, Head form: Manometers have historically lead to a desire to think of energy in units of length, or head. We see that we can arrive at units of length by dividing our velocity squared form of the energy equation by gravity. Hence we have: û out g + v2 out 2g + z out + out γ = ûin g + v2 in 2g + z in + in γ + h q h s where h q and h s are q/g and w s /g, respectively the head forms of the heat transfer power and shaft power Incompressible 1-D Flow With No Shaft Work Rearranging our 1-D head form and setting w s =0we have: ( ) ( ) 2g + z = out 2g + z ûout û in q in g Now, defining û out û in q g = h loss = h f where we can think of h f as the friction losses and we see that h f > 0 (note that in adiabatic flow, say a perfectly insulated pipe, friction will heat the flow, therefore û out > û in and hence h f > 0). Thus we write ( ) ( ) 2g + z = out 2g + z h f in

6 Example Gas ipeline Consider the following pipe flow: If Q = 75 m 3 /s, the pipe radius is r =6cm, the inlet pressure is maintained at 24 atm by a pump, the outlet vents to the atmosphere, the pipe rises 150 m from inlet to outlet and the pipe length is 10 km, what is h f? What is the velocity head? h f =198 m Therefore the friction loss is greater than the z and the pump must drive against both! The velocity head is only 0.17 m! Note that the length did not come into our solution. h f includes the total losses along the pipe due to friction effects and hence includes the effect of length implicitly. We will see more about this a bit later in the course.

7 CEE 3310 Control Volume Analysis, Oct. 3, Review Work and ower The Energy Equation Q Ẇshaft = d ˆ dt CV ) ˆ (û + v2 2 + gz d + (û + v2 CS 2 + gz + ) ( v n) da where Q is the heat energy transfer rate, Ẇ shaft is the shaft power (work rate), û is the internal energy per unit mass, v 2 /2 is the kinetic energy per unit mass, and gz is the potential energy per unit mass Bernoulli Equation Bernoulli wrote down a verbal form of his famous equation in 1738 and Euler completed the analytic derivation in The differential form of the Bernoulli Equation is known as the Euler Equation. Consider our 1-D head form of the energy equation and let s apply it along a streamline of a flow. If the flow is steady then the integral over the control volume vanishes. Further, since by definition there is no flow normal to the streamline we only have flux terms at the starting and ending points along the streamline (really we are talking about a volume and hence a streamtube just a cylindrical volume element defined by a family of streamlines - a virtual pipe!). Clearly there is no shaft work along the streamline. If we assume the flow is frictionless (i.e., inviscid or ν =0) h f =0 and we have ( ) ( ) 2g + z = 2g + z out This is the Bernoulli equation. Clearly anywhere along a streamline, as long as no work is done between analysis points and the assumption of frictionless flow is good, we can write 2g + z = h 0 in

8 72 where the constant h 0 is referred to as the Bernoulli constant and varies across streamlines. The Bernoulli Equation can be derived by considering Newton s second law F = m a along a streamline (conservation of linear momentum). This leads to the steady form of the Bernoulli equation. If we add conservation of mass we can derive the unsteady form. Bernoulli Equation Assumptions Flow along single streamline different streamlines, different h 0. Steady flow (can be generalized to unsteady flow). Incompressible flow. Inviscid or frictionless flow, very restrictive! No w s between analysis points on streamline. No q between points on streamline Illustrations of Valid and Invalid Regions for the Application of the Bernoulli Equation

9 CEE 3310 Control Volume Analysis, Oct. 3, ressure form of Bernoulli Equation If we multiply our head form of the Bernoulli equation by the specific weight we arrive at the pressure form of the Bernoulli Equation: + v2 2 + γz = t where we call the first term the static pressure, the second the dynamic pressure, the third the hydrostatic pressure, and the right-hand-side the total pressure. Hence the Bernoulli Equation says that in inviscid flows the total pressure along a streamline is constant. If we remain at a constant elevation the above equation reduces to + v2 2 = s where we refer to s as the stagnation pressure. Thus by definition the stagnation pressure is the pressure along horizontal streamlines when the velocity is zero Stagnation oint and ressure Consider the flow around a circular cylinder: + v2 2 + γz = t We see that the stagnation pressure is simply the conversion of all kinetic energy to potential energy and hence there is a subsequent pressure rise. The elevation head simply accounts for any change in the potential energy due to vertical changes in elevation.

10 itot-static Tube The static and stagnation pressures can be measured simultaneously using a itot-static tube. Consider the following geometry: Now, we see the streamlines around the tube, either at the tip or away from the tip (but not around the curved front end), are horizontal. If the tube is not very long it is very reasonable to assume friction is negligible for this analysis. Now the velocity at the tip of the itot-static tube is zero hence the pressure at this point (and hence along the entire horizontal leg of the itot tube as this portion of the device is known) is the stagnation pressure. The holes perpendicular to the flow are similar to piezometers - they simply measure the static pressure in the fluid flow. If we write the equation for the itot tube we have 1 + v2 1 2 = 2 = γh Now, at the static tube we have the free-stream pressure, 2o but at this point, which is along the streamline from point 1 to point 2, the velocity is the same as it is for point 1 (assuming the itot-static tube is small and does not affect the flow) and there is no elevation change so the Bernoulli Equation gives us: 1 = 2o = γh Substituting this expression into the equation for the itot tube we arrive at the itot formula or in terms of heads V 1 = V = 2g(H h)

11 CEE 3310 Control Volume Analysis, Oct. 3, Example Flow accelerating out of a reservoir 2gh V 2 = 1 ( ) 2 A2 A and if A 1 A 2 : 1 ( A2 A 1 ) 2 1 V 2 2gh, again! Let s look at this a bit further by asking the question what speed will a parcel of fluid dropped a distance h be traveling at? Therefore t = v = ˆ t 0 2h g g dt = gt h = ˆ t and v = gt v = g 0 v dt = ˆ t 0 gt dt = 1 2 gt2 2h g = 2gh Thus we see that in an inviscid flow, which by definition has no frictional energy losses, we simply convert potential energy to kinetic energy and hence the same result, v = 2gh keeps showing up. This was first noted by Torricelli. aha!

12 Irrotational Flow and Bernoulli Consider h 4 γ γ 2g + z = h 0 0+ v2 2g + 0 = h 0 = v2 2g h 2 γ γ + v2 2g h 2 = h 0 h 3 γ γ + v2 2g h 3 = h 0 v 2 + (v/2)2 h 4 = 1 2g 4 2g h 0 h 5 γ γ +0 h 5 = 0 h 0 Notice that in the unsheared regions (uniform flow) h 0 = a constant across streamlines while where shear exists (e.g., shear is non-zero), h 0 varies across the streamlines. More strictly speaking we actually want to know if the flow is rotational. Our test is if we stick a small neutrally buoyant + shaped probe in the flow and see if it will rotate. In uniform flow it will not, in a linear shear, like the shear profile shown here, it will. Hence we say that h 0 is constant in irrotational (non rotational) flows. This allows us to connect Bernoulli points that are not on the same streamline in flows that are irrotational, further expanding the power of the Bernoulli equation but also the opportunities to misuse it!

CEE 3310 Control Volume Analysis, Oct. 10, = dt. sys

CEE 3310 Control Volume Analysis, Oct. 10, = dt. sys CEE 3310 Control Volume Analysis, Oct. 10, 2018 77 3.16 Review First Law of Thermodynamics ( ) de = dt Q Ẇ sys Sign convention: Work done by the surroundings on the system < 0, example, a pump! Work done

More information

3.25 Pressure form of Bernoulli Equation

3.25 Pressure form of Bernoulli Equation CEE 3310 Control Volume Analysis, Oct 3, 2012 83 3.24 Review The Energy Equation Q Ẇshaft = d dt CV ) (û + v2 2 + gz ρ d + (û + v2 CS 2 + gz + ) ρ( v n) da ρ where Q is the heat energy transfer rate, Ẇ

More information

CEE 3310 Control Volume Analysis, Oct. 7, D Steady State Head Form of the Energy Equation P. P 2g + z h f + h p h s.

CEE 3310 Control Volume Analysis, Oct. 7, D Steady State Head Form of the Energy Equation P. P 2g + z h f + h p h s. CEE 3310 Control Volume Analysis, Oct. 7, 2015 81 3.21 Review 1-D Steady State Head Form of the Energy Equation ( ) ( ) 2g + z = 2g + z h f + h p h s out where h f is the friction head loss (which combines

More information

Objectives. Conservation of mass principle: Mass Equation The Bernoulli equation Conservation of energy principle: Energy equation

Objectives. Conservation of mass principle: Mass Equation The Bernoulli equation Conservation of energy principle: Energy equation Objectives Conservation of mass principle: Mass Equation The Bernoulli equation Conservation of energy principle: Energy equation Conservation of Mass Conservation of Mass Mass, like energy, is a conserved

More information

vector H. If O is the point about which moments are desired, the angular moment about O is given:

vector H. If O is the point about which moments are desired, the angular moment about O is given: The angular momentum A control volume analysis can be applied to the angular momentum, by letting B equal to angularmomentum vector H. If O is the point about which moments are desired, the angular moment

More information

Angular momentum equation

Angular momentum equation Angular momentum equation For angular momentum equation, B =H O the angular momentum vector about point O which moments are desired. Where β is The Reynolds transport equation can be written as follows:

More information

Chapter Four fluid flow mass, energy, Bernoulli and momentum

Chapter Four fluid flow mass, energy, Bernoulli and momentum 4-1Conservation of Mass Principle Consider a control volume of arbitrary shape, as shown in Fig (4-1). Figure (4-1): the differential control volume and differential control volume (Total mass entering

More information

Mass of fluid leaving per unit time

Mass of fluid leaving per unit time 5 ENERGY EQUATION OF FLUID MOTION 5.1 Eulerian Approach & Control Volume In order to develop the equations that describe a flow, it is assumed that fluids are subject to certain fundamental laws of physics.

More information

5 ENERGY EQUATION OF FLUID MOTION

5 ENERGY EQUATION OF FLUID MOTION 5 ENERGY EQUATION OF FLUID MOTION 5.1 Introduction In order to develop the equations that describe a flow, it is assumed that fluids are subject to certain fundamental laws of physics. The pertinent laws

More information

Lesson 6 Review of fundamentals: Fluid flow

Lesson 6 Review of fundamentals: Fluid flow Lesson 6 Review of fundamentals: Fluid flow The specific objective of this lesson is to conduct a brief review of the fundamentals of fluid flow and present: A general equation for conservation of mass

More information

Chapter 5: Mass, Bernoulli, and Energy Equations

Chapter 5: Mass, Bernoulli, and Energy Equations Chapter 5: Mass, Bernoulli, and Energy Equations Introduction This chapter deals with 3 equations commonly used in fluid mechanics The mass equation is an expression of the conservation of mass principle.

More information

Basic Fluid Mechanics

Basic Fluid Mechanics Basic Fluid Mechanics Chapter 5: Application of Bernoulli Equation 4/16/2018 C5: Application of Bernoulli Equation 1 5.1 Introduction In this chapter we will show that the equation of motion of a particle

More information

Chapter 7 The Energy Equation

Chapter 7 The Energy Equation Chapter 7 The Energy Equation 7.1 Energy, Work, and Power When matter has energy, the matter can be used to do work. A fluid can have several forms of energy. For example a fluid jet has kinetic energy,

More information

MASS, MOMENTUM, AND ENERGY EQUATIONS

MASS, MOMENTUM, AND ENERGY EQUATIONS MASS, MOMENTUM, AND ENERGY EQUATIONS This chapter deals with four equations commonly used in fluid mechanics: the mass, Bernoulli, Momentum and energy equations. The mass equation is an expression of the

More information

SYSTEMS VS. CONTROL VOLUMES. Control volume CV (open system): Arbitrary geometric space, surrounded by control surfaces (CS)

SYSTEMS VS. CONTROL VOLUMES. Control volume CV (open system): Arbitrary geometric space, surrounded by control surfaces (CS) SYSTEMS VS. CONTROL VOLUMES System (closed system): Predefined mass m, surrounded by a system boundary Control volume CV (open system): Arbitrary geometric space, surrounded by control surfaces (CS) Many

More information

Where does Bernoulli's Equation come from?

Where does Bernoulli's Equation come from? Where does Bernoulli's Equation come from? Introduction By now, you have seen the following equation many times, using it to solve simple fluid problems. P ρ + v + gz = constant (along a streamline) This

More information

where = rate of change of total energy of the system, = rate of heat added to the system, = rate of work done by the system

where = rate of change of total energy of the system, = rate of heat added to the system, = rate of work done by the system The Energy Equation for Control Volumes Recall, the First Law of Thermodynamics: where = rate of change of total energy of the system, = rate of heat added to the system, = rate of work done by the system

More information

Chapter 5: Mass, Bernoulli, and

Chapter 5: Mass, Bernoulli, and and Energy Equations 5-1 Introduction 5-2 Conservation of Mass 5-3 Mechanical Energy 5-4 General Energy Equation 5-5 Energy Analysis of Steady Flows 5-6 The Bernoulli Equation 5-1 Introduction This chapter

More information

Chapter 3 Bernoulli Equation

Chapter 3 Bernoulli Equation 1 Bernoulli Equation 3.1 Flow Patterns: Streamlines, Pathlines, Streaklines 1) A streamline, is a line that is everywhere tangent to the velocity vector at a given instant. Examples of streamlines around

More information

Chapter 2: Basic Governing Equations

Chapter 2: Basic Governing Equations -1 Reynolds Transport Theorem (RTT) - Continuity Equation -3 The Linear Momentum Equation -4 The First Law of Thermodynamics -5 General Equation in Conservative Form -6 General Equation in Non-Conservative

More information

CLASS SCHEDULE 2013 FALL

CLASS SCHEDULE 2013 FALL CLASS SCHEDULE 2013 FALL Class # or Lab # 1 Date Aug 26 2 28 Important Concepts (Section # in Text Reading, Lecture note) Examples/Lab Activities Definition fluid; continuum hypothesis; fluid properties

More information

10.52 Mechanics of Fluids Spring 2006 Problem Set 3

10.52 Mechanics of Fluids Spring 2006 Problem Set 3 10.52 Mechanics of Fluids Spring 2006 Problem Set 3 Problem 1 Mass transfer studies involving the transport of a solute from a gas to a liquid often involve the use of a laminar jet of liquid. The situation

More information

Introduction to Turbomachinery

Introduction to Turbomachinery 1. Coordinate System Introduction to Turbomachinery Since there are stationary and rotating blades in turbomachines, they tend to form a cylindrical form, represented in three directions; 1. Axial 2. Radial

More information

Shell/Integral Balances (SIB)

Shell/Integral Balances (SIB) Shell/Integral Balances (SIB) Shell/Integral Balances Shell or integral (macroscopic) balances are often relatively simple to solve, both conceptually and mechanically, as only limited data is necessary.

More information

Unit C-1: List of Subjects

Unit C-1: List of Subjects Unit C-: List of Subjects The elocity Field The Acceleration Field The Material or Substantial Derivative Steady Flow and Streamlines Fluid Particle in a Flow Field F=ma along a Streamline Bernoulli s

More information

MAE 3130: Fluid Mechanics Lecture 7: Differential Analysis/Part 1 Spring Dr. Jason Roney Mechanical and Aerospace Engineering

MAE 3130: Fluid Mechanics Lecture 7: Differential Analysis/Part 1 Spring Dr. Jason Roney Mechanical and Aerospace Engineering MAE 3130: Fluid Mechanics Lecture 7: Differential Analysis/Part 1 Spring 2003 Dr. Jason Roney Mechanical and Aerospace Engineering Outline Introduction Kinematics Review Conservation of Mass Stream Function

More information

Aerodynamics. Basic Aerodynamics. Continuity equation (mass conserved) Some thermodynamics. Energy equation (energy conserved)

Aerodynamics. Basic Aerodynamics. Continuity equation (mass conserved) Some thermodynamics. Energy equation (energy conserved) Flow with no friction (inviscid) Aerodynamics Basic Aerodynamics Continuity equation (mass conserved) Flow with friction (viscous) Momentum equation (F = ma) 1. Euler s equation 2. Bernoulli s equation

More information

FE Fluids Review March 23, 2012 Steve Burian (Civil & Environmental Engineering)

FE Fluids Review March 23, 2012 Steve Burian (Civil & Environmental Engineering) Topic: Fluid Properties 1. If 6 m 3 of oil weighs 47 kn, calculate its specific weight, density, and specific gravity. 2. 10.0 L of an incompressible liquid exert a force of 20 N at the earth s surface.

More information

The Bernoulli Equation

The Bernoulli Equation The Bernoulli Equation The most used and the most abused equation in fluid mechanics. Newton s Second Law: F = ma In general, most real flows are 3-D, unsteady (x, y, z, t; r,θ, z, t; etc) Let consider

More information

Chapter 14. Lecture 1 Fluid Mechanics. Dr. Armen Kocharian

Chapter 14. Lecture 1 Fluid Mechanics. Dr. Armen Kocharian Chapter 14 Lecture 1 Fluid Mechanics Dr. Armen Kocharian States of Matter Solid Has a definite volume and shape Liquid Has a definite volume but not a definite shape Gas unconfined Has neither a definite

More information

4 Mechanics of Fluids (I)

4 Mechanics of Fluids (I) 1. The x and y components of velocity for a two-dimensional flow are u = 3.0 ft/s and v = 9.0x ft/s where x is in feet. Determine the equation for the streamlines and graph representative streamlines in

More information

Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015

Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015 Detailed Outline, M E 320 Fluid Flow, Spring Semester 2015 I. Introduction (Chapters 1 and 2) A. What is Fluid Mechanics? 1. What is a fluid? 2. What is mechanics? B. Classification of Fluid Flows 1. Viscous

More information

Fluid Mechanics-61341

Fluid Mechanics-61341 An-Najah National University College of Engineering Fluid Mechanics-61341 Chapter [5] Flow of An Incompressible Fluid Dr. Sameer Shadeed 1 Fluid Mechanics-2nd Semester 2010- [5] Flow of An Incompressible

More information

Stream Tube. When density do not depend explicitly on time then from continuity equation, we have V 2 V 1. δa 2. δa 1 PH6L24 1

Stream Tube. When density do not depend explicitly on time then from continuity equation, we have V 2 V 1. δa 2. δa 1 PH6L24 1 Stream Tube A region of the moving fluid bounded on the all sides by streamlines is called a tube of flow or stream tube. As streamline does not intersect each other, no fluid enters or leaves across the

More information

Dimensions represent classes of units we use to describe a physical quantity. Most fluid problems involve four primary dimensions

Dimensions represent classes of units we use to describe a physical quantity. Most fluid problems involve four primary dimensions BEE 5330 Fluids FE Review, Feb 24, 2010 1 A fluid is a substance that can not support a shear stress. Liquids differ from gasses in that liquids that do not completely fill a container will form a free

More information

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t) IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common

More information

EGN 3353C Fluid Mechanics

EGN 3353C Fluid Mechanics Lecture 8 Bernoulli s Equation: Limitations and Applications Last time, we derived the steady form of Bernoulli s Equation along a streamline p + ρv + ρgz = P t static hydrostatic total pressure q = dynamic

More information

CHAPTER 3 BASIC EQUATIONS IN FLUID MECHANICS NOOR ALIZA AHMAD

CHAPTER 3 BASIC EQUATIONS IN FLUID MECHANICS NOOR ALIZA AHMAD CHAPTER 3 BASIC EQUATIONS IN FLUID MECHANICS 1 INTRODUCTION Flow often referred as an ideal fluid. We presume that such a fluid has no viscosity. However, this is an idealized situation that does not exist.

More information

Lagrangian description from the perspective of a parcel moving within the flow. Streamline Eulerian, tangent line to instantaneous velocity field.

Lagrangian description from the perspective of a parcel moving within the flow. Streamline Eulerian, tangent line to instantaneous velocity field. Chapter 2 Hydrostatics 2.1 Review Eulerian description from the perspective of fixed points within a reference frame. Lagrangian description from the perspective of a parcel moving within the flow. Streamline

More information

Fundamentals of Fluid Mechanics

Fundamentals of Fluid Mechanics Sixth Edition Fundamentals of Fluid Mechanics International Student Version BRUCE R. MUNSON DONALD F. YOUNG Department of Aerospace Engineering and Engineering Mechanics THEODORE H. OKIISHI Department

More information

ME3560 Tentative Schedule Fall 2018

ME3560 Tentative Schedule Fall 2018 ME3560 Tentative Schedule Fall 2018 Week Number 1 Wednesday 8/29/2018 1 Date Lecture Topics Covered Introduction to course, syllabus and class policies. Math Review. Differentiation. Prior to Lecture Read

More information

Basic Fluid Mechanics

Basic Fluid Mechanics Basic Fluid Mechanics Chapter 6A: Internal Incompressible Viscous Flow 4/16/2018 C6A: Internal Incompressible Viscous Flow 1 6.1 Introduction For the present chapter we will limit our study to incompressible

More information

CLASS Fourth Units (Second part)

CLASS Fourth Units (Second part) CLASS Fourth Units (Second part) Energy analysis of closed systems Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. MOVING BOUNDARY WORK Moving boundary work (P

More information

Chapter 14. Fluid Mechanics

Chapter 14. Fluid Mechanics Chapter 14 Fluid Mechanics States of Matter Solid Has a definite volume and shape Liquid Has a definite volume but not a definite shape Gas unconfined Has neither a definite volume nor shape All of these

More information

ME3560 Tentative Schedule Spring 2019

ME3560 Tentative Schedule Spring 2019 ME3560 Tentative Schedule Spring 2019 Week Number Date Lecture Topics Covered Prior to Lecture Read Section Assignment Prep Problems for Prep Probs. Must be Solved by 1 Monday 1/7/2019 1 Introduction to

More information

150A Review Session 2/13/2014 Fluid Statics. Pressure acts in all directions, normal to the surrounding surfaces

150A Review Session 2/13/2014 Fluid Statics. Pressure acts in all directions, normal to the surrounding surfaces Fluid Statics Pressure acts in all directions, normal to the surrounding surfaces or Whenever a pressure difference is the driving force, use gauge pressure o Bernoulli equation o Momentum balance with

More information

Chapter 6: Momentum Analysis

Chapter 6: Momentum Analysis 6-1 Introduction 6-2Newton s Law and Conservation of Momentum 6-3 Choosing a Control Volume 6-4 Forces Acting on a Control Volume 6-5Linear Momentum Equation 6-6 Angular Momentum 6-7 The Second Law of

More information

Fluid Mechanics. du dy

Fluid Mechanics. du dy FLUID MECHANICS Technical English - I 1 th week Fluid Mechanics FLUID STATICS FLUID DYNAMICS Fluid Statics or Hydrostatics is the study of fluids at rest. The main equation required for this is Newton's

More information

Conservation of Momentum using Control Volumes

Conservation of Momentum using Control Volumes Conservation of Momentum using Control Volumes Conservation of Linear Momentum Recall the conservation of linear momentum law for a system: In order to convert this for use in a control volume, use RTT

More information

FLUID MECHANICS. Chapter 3 Elementary Fluid Dynamics - The Bernoulli Equation

FLUID MECHANICS. Chapter 3 Elementary Fluid Dynamics - The Bernoulli Equation FLUID MECHANICS Chapter 3 Elementary Fluid Dynamics - The Bernoulli Equation CHAP 3. ELEMENTARY FLUID DYNAMICS - THE BERNOULLI EQUATION CONTENTS 3. Newton s Second Law 3. F = ma along a Streamline 3.3

More information

Exam #2: Fluid Kinematics and Conservation Laws April 13, 2016, 7:00 p.m. 8:40 p.m. in CE 118

Exam #2: Fluid Kinematics and Conservation Laws April 13, 2016, 7:00 p.m. 8:40 p.m. in CE 118 CVEN 311-501 (Socolofsky) Fluid Dynamics Exam #2: Fluid Kinematics and Conservation Laws April 13, 2016, 7:00 p.m. 8:40 p.m. in CE 118 Name: : UIN: : Instructions: Fill in your name and UIN in the space

More information

Chapter Two. Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency. Laith Batarseh

Chapter Two. Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency. Laith Batarseh Chapter Two Basic Thermodynamics, Fluid Mechanics: Definitions of Efficiency Laith Batarseh The equation of continuity Most analyses in this book are limited to one-dimensional steady flows where the velocity

More information

Useful concepts associated with the Bernoulli equation. Dynamic

Useful concepts associated with the Bernoulli equation. Dynamic Useful concets associated with the Bernoulli equation - Static, Stagnation, and Dynamic Pressures Bernoulli eq. along a streamline + ρ v + γ z = constant (Unit of Pressure Static (Thermodynamic Dynamic

More information

2.The lines that are tangent to the velocity vectors throughout the flow field are called steady flow lines. True or False A. True B.

2.The lines that are tangent to the velocity vectors throughout the flow field are called steady flow lines. True or False A. True B. CHAPTER 03 1. Write Newton's second law of motion. YOUR ANSWER: F = ma 2.The lines that are tangent to the velocity vectors throughout the flow field are called steady flow lines. True or False 3.Streamwise

More information

Part A: 1 pts each, 10 pts total, no partial credit.

Part A: 1 pts each, 10 pts total, no partial credit. Part A: 1 pts each, 10 pts total, no partial credit. 1) (Correct: 1 pt/ Wrong: -3 pts). The sum of static, dynamic, and hydrostatic pressures is constant when flow is steady, irrotational, incompressible,

More information

In this section, mathematical description of the motion of fluid elements moving in a flow field is

In this section, mathematical description of the motion of fluid elements moving in a flow field is Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small

More information

!! +! 2!! +!"!! =!! +! 2!! +!"!! +!!"!"!"

!! +! 2!! +!!! =!! +! 2!! +!!! +!!!! Homework 4 Solutions 1. (15 points) Bernoulli s equation can be adapted for use in evaluating unsteady flow conditions, such as those encountered during start- up processes. For example, consider the large

More information

BERNOULLI EQUATION. The motion of a fluid is usually extremely complex.

BERNOULLI EQUATION. The motion of a fluid is usually extremely complex. BERNOULLI EQUATION The motion of a fluid is usually extremely complex. The study of a fluid at rest, or in relative equilibrium, was simplified by the absence of shear stress, but when a fluid flows over

More information

Lecture 4. Differential Analysis of Fluid Flow Navier-Stockes equation

Lecture 4. Differential Analysis of Fluid Flow Navier-Stockes equation Lecture 4 Differential Analysis of Fluid Flow Navier-Stockes equation Newton second law and conservation of momentum & momentum-of-momentum A jet of fluid deflected by an object puts a force on the object.

More information

ROAD MAP... D-0: Reynolds Transport Theorem D-1: Conservation of Mass D-2: Conservation of Momentum D-3: Conservation of Energy

ROAD MAP... D-0: Reynolds Transport Theorem D-1: Conservation of Mass D-2: Conservation of Momentum D-3: Conservation of Energy ES06 Fluid Mechani UNIT D: Flow Field Analysis ROAD MAP... D-0: Reynolds Transport Theorem D-1: Conservation of Mass D-: Conservation of Momentum D-3: Conservation of Energy ES06 Fluid Mechani Unit D-0:

More information

Chapter 4 DYNAMICS OF FLUID FLOW

Chapter 4 DYNAMICS OF FLUID FLOW Faculty Of Engineering at Shobra nd Year Civil - 016 Chapter 4 DYNAMICS OF FLUID FLOW 4-1 Types of Energy 4- Euler s Equation 4-3 Bernoulli s Equation 4-4 Total Energy Line (TEL) and Hydraulic Grade Line

More information

COURSE NUMBER: ME 321 Fluid Mechanics I 3 credit hour. Basic Equations in fluid Dynamics

COURSE NUMBER: ME 321 Fluid Mechanics I 3 credit hour. Basic Equations in fluid Dynamics COURSE NUMBER: ME 321 Fluid Mechanics I 3 credit hour Basic Equations in fluid Dynamics Course teacher Dr. M. Mahbubur Razzaque Professor Department of Mechanical Engineering BUET 1 Description of Fluid

More information

Review of fluid dynamics

Review of fluid dynamics Chapter 2 Review of fluid dynamics 2.1 Preliminaries ome basic concepts: A fluid is a substance that deforms continuously under stress. A Material olume is a tagged region that moves with the fluid. Hence

More information

Differential relations for fluid flow

Differential relations for fluid flow Differential relations for fluid flow In this approach, we apply basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of a flow

More information

MOMENTUM PRINCIPLE. Review: Last time, we derived the Reynolds Transport Theorem: Chapter 6. where B is any extensive property (proportional to mass),

MOMENTUM PRINCIPLE. Review: Last time, we derived the Reynolds Transport Theorem: Chapter 6. where B is any extensive property (proportional to mass), Chapter 6 MOMENTUM PRINCIPLE Review: Last time, we derived the Reynolds Transport Theorem: where B is any extensive property (proportional to mass), and b is the corresponding intensive property (B / m

More information

Lecture 3 The energy equation

Lecture 3 The energy equation Lecture 3 The energy equation Dr Tim Gough: t.gough@bradford.ac.uk General information Lab groups now assigned Timetable up to week 6 published Is there anyone not yet on the list? Week 3 Week 4 Week 5

More information

For example an empty bucket weighs 2.0kg. After 7 seconds of collecting water the bucket weighs 8.0kg, then:

For example an empty bucket weighs 2.0kg. After 7 seconds of collecting water the bucket weighs 8.0kg, then: Hydraulic Coefficient & Flow Measurements ELEMENTARY HYDRAULICS National Certificate in Technology (Civil Engineering) Chapter 3 1. Mass flow rate If we want to measure the rate at which water is flowing

More information

first law of ThermodyNamics

first law of ThermodyNamics first law of ThermodyNamics First law of thermodynamics - Principle of conservation of energy - Energy can be neither created nor destroyed Basic statement When any closed system is taken through a cycle,

More information

ENERGY TRANSFER BETWEEN FLUID AND ROTOR. Dr. Ir. Harinaldi, M.Eng Mechanical Engineering Department Faculty of Engineering University of Indonesia

ENERGY TRANSFER BETWEEN FLUID AND ROTOR. Dr. Ir. Harinaldi, M.Eng Mechanical Engineering Department Faculty of Engineering University of Indonesia ENERGY TRANSFER BETWEEN FLUID AND ROTOR Dr. Ir. Harinaldi, M.Eng Mechanical Engineering Department Faculty of Engineering University of Indonesia Basic Laws and Equations Continuity Equation m m ρ mass

More information

Basics of fluid flow. Types of flow. Fluid Ideal/Real Compressible/Incompressible

Basics of fluid flow. Types of flow. Fluid Ideal/Real Compressible/Incompressible Basics of fluid flow Types of flow Fluid Ideal/Real Compressible/Incompressible Flow Steady/Unsteady Uniform/Non-uniform Laminar/Turbulent Pressure/Gravity (free surface) 1 Basics of fluid flow (Chapter

More information

11.1 Mass Density. Fluids are materials that can flow, and they include both gases and liquids. The mass density of a liquid or gas is an

11.1 Mass Density. Fluids are materials that can flow, and they include both gases and liquids. The mass density of a liquid or gas is an Chapter 11 Fluids 11.1 Mass Density Fluids are materials that can flow, and they include both gases and liquids. The mass density of a liquid or gas is an important factor that determines its behavior

More information

V/ t = 0 p/ t = 0 ρ/ t = 0. V/ s = 0 p/ s = 0 ρ/ s = 0

V/ t = 0 p/ t = 0 ρ/ t = 0. V/ s = 0 p/ s = 0 ρ/ s = 0 UNIT III FLOW THROUGH PIPES 1. List the types of fluid flow. Steady and unsteady flow Uniform and non-uniform flow Laminar and Turbulent flow Compressible and incompressible flow Rotational and ir-rotational

More information

Introduction to Aerodynamics. Dr. Guven Aerospace Engineer (P.hD)

Introduction to Aerodynamics. Dr. Guven Aerospace Engineer (P.hD) Introduction to Aerodynamics Dr. Guven Aerospace Engineer (P.hD) Aerodynamic Forces All aerodynamic forces are generated wither through pressure distribution or a shear stress distribution on a body. The

More information

Theory of turbomachinery. Chapter 1

Theory of turbomachinery. Chapter 1 Theory of turbomachinery Chater Introduction: Basic Princiles Take your choice of those that can best aid your action. (Shakeseare, Coriolanus) Introduction Definition Turbomachinery describes machines

More information

Iran University of Science & Technology School of Mechanical Engineering Advance Fluid Mechanics

Iran University of Science & Technology School of Mechanical Engineering Advance Fluid Mechanics 1. Consider a sphere of radius R immersed in a uniform stream U0, as shown in 3 R Fig.1. The fluid velocity along streamline AB is given by V ui U i x 1. 0 3 Find (a) the position of maximum fluid acceleration

More information

6.1 Momentum Equation for Frictionless Flow: Euler s Equation The equations of motion for frictionless flow, called Euler s

6.1 Momentum Equation for Frictionless Flow: Euler s Equation The equations of motion for frictionless flow, called Euler s Chapter 6 INCOMPRESSIBLE INVISCID FLOW All real fluids possess viscosity. However in many flow cases it is reasonable to neglect the effects of viscosity. It is useful to investigate the dynamics of an

More information

AA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow

AA210A Fundamentals of Compressible Flow. Chapter 1 - Introduction to fluid flow AA210A Fundamentals of Compressible Flow Chapter 1 - Introduction to fluid flow 1 1.2 Conservation of mass Mass flux in the x-direction [ ρu ] = M L 3 L T = M L 2 T Momentum per unit volume Mass per unit

More information

Answers to questions in each section should be tied together and handed in separately.

Answers to questions in each section should be tied together and handed in separately. EGT0 ENGINEERING TRIPOS PART IA Wednesday 4 June 014 9 to 1 Paper 1 MECHANICAL ENGINEERING Answer all questions. The approximate number of marks allocated to each part of a question is indicated in the

More information

Fluid Mechanics. Spring 2009

Fluid Mechanics. Spring 2009 Instructor: Dr. Yang-Cheng Shih Department of Energy and Refrigerating Air-Conditioning Engineering National Taipei University of Technology Spring 2009 Chapter 1 Introduction 1-1 General Remarks 1-2 Scope

More information

Applied Fluid Mechanics

Applied Fluid Mechanics Applied Fluid Mechanics 1. The Nature of Fluid and the Study of Fluid Mechanics 2. Viscosity of Fluid 3. Pressure Measurement 4. Forces Due to Static Fluid 5. Buoyancy and Stability 6. Flow of Fluid and

More information

ME332 FLUID MECHANICS LABORATORY (PART II)

ME332 FLUID MECHANICS LABORATORY (PART II) ME332 FLUID MECHANICS LABORATORY (PART II) Mihir Sen Department of Aerospace and Mechanical Engineering University of Notre Dame Notre Dame, IN 46556 Version: April 2, 2002 Contents Unit 5: Momentum transfer

More information

Physics 3 Summer 1990 Lab 7 - Hydrodynamics

Physics 3 Summer 1990 Lab 7 - Hydrodynamics Physics 3 Summer 1990 Lab 7 - Hydrodynamics Theory Consider an ideal liquid, one which is incompressible and which has no internal friction, flowing through pipe of varying cross section as shown in figure

More information

Heat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay

Heat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay Heat and Mass Transfer Prof. S.P. Sukhatme Department of Mechanical Engineering Indian Institute of Technology, Bombay Lecture No. 18 Forced Convection-1 Welcome. We now begin our study of forced convection

More information

CHEN 3200 Fluid Mechanics Spring Homework 3 solutions

CHEN 3200 Fluid Mechanics Spring Homework 3 solutions Homework 3 solutions 1. An artery with an inner diameter of 15 mm contains blood flowing at a rate of 5000 ml/min. Further along the artery, arterial plaque has partially clogged the artery, reducing the

More information

Energy CEEN 598D: Fluid Mechanics for Hydro Systems. Lindsay Bearup Berthoud Hall 121

Energy CEEN 598D: Fluid Mechanics for Hydro Systems. Lindsay Bearup Berthoud Hall 121 Energy CEEN 598D: Fluid Mechanics for Hydro Systems Lindsay Bearup lbearup@mines.edu Berthoud Hall 11 GEGN 498A Fall 013 For the record: Material DerivaIves db sys dt Rate of change of property B of system

More information

Basic Thermodynamics Prof. S.K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur

Basic Thermodynamics Prof. S.K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Basic Thermodynamics Prof. S.K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture 12 Second Law and Available Energy III Good morning. I welcome you to this session.

More information

Chapter 5. The Differential Forms of the Fundamental Laws

Chapter 5. The Differential Forms of the Fundamental Laws Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations

More information

Visualization of flow pattern over or around immersed objects in open channel flow.

Visualization of flow pattern over or around immersed objects in open channel flow. EXPERIMENT SEVEN: FLOW VISUALIZATION AND ANALYSIS I OBJECTIVE OF THE EXPERIMENT: Visualization of flow pattern over or around immersed objects in open channel flow. II THEORY AND EQUATION: Open channel:

More information

Introduction to Chemical Engineering Thermodynamics. Chapter 7. KFUPM Housam Binous CHE 303

Introduction to Chemical Engineering Thermodynamics. Chapter 7. KFUPM Housam Binous CHE 303 Introduction to Chemical Engineering Thermodynamics Chapter 7 1 Thermodynamics of flow is based on mass, energy and entropy balances Fluid mechanics encompasses the above balances and conservation of momentum

More information

Pressure in stationary and moving fluid Lab- Lab On- On Chip: Lecture 2

Pressure in stationary and moving fluid Lab- Lab On- On Chip: Lecture 2 Pressure in stationary and moving fluid Lab-On-Chip: Lecture Lecture plan what is pressure e and how it s distributed in static fluid water pressure in engineering problems buoyancy y and archimedes law;

More information

If a stream of uniform velocity flows into a blunt body, the stream lines take a pattern similar to this: Streamlines around a blunt body

If a stream of uniform velocity flows into a blunt body, the stream lines take a pattern similar to this: Streamlines around a blunt body Venturimeter & Orificemeter ELEMENTARY HYDRAULICS National Certificate in Technology (Civil Engineering) Chapter 5 Applications of the Bernoulli Equation The Bernoulli equation can be applied to a great

More information

TOPICS. Density. Pressure. Variation of Pressure with Depth. Pressure Measurements. Buoyant Forces-Archimedes Principle

TOPICS. Density. Pressure. Variation of Pressure with Depth. Pressure Measurements. Buoyant Forces-Archimedes Principle Lecture 6 Fluids TOPICS Density Pressure Variation of Pressure with Depth Pressure Measurements Buoyant Forces-Archimedes Principle Surface Tension ( External source ) Viscosity ( External source ) Equation

More information

Introduction to Fluid Machines, and Compressible Flow Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur

Introduction to Fluid Machines, and Compressible Flow Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Introduction to Fluid Machines, and Compressible Flow Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture - 09 Introduction to Reaction Type of Hydraulic

More information

CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION

CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION CHAPTER 7 SEVERAL FORMS OF THE EQUATIONS OF MOTION 7.1 THE NAVIER-STOKES EQUATIONS Under the assumption of a Newtonian stress-rate-of-strain constitutive equation and a linear, thermally conductive medium,

More information

AER210 VECTOR CALCULUS and FLUID MECHANICS. Quiz 4 Duration: 70 minutes

AER210 VECTOR CALCULUS and FLUID MECHANICS. Quiz 4 Duration: 70 minutes AER210 VECTOR CALCULUS and FLUID MECHANICS Quiz 4 Duration: 70 minutes 26 November 2012 Closed Book, no aid sheets Non-programmable calculators allowed Instructor: Alis Ekmekci Family Name: Given Name:

More information

Sourabh V. Apte. 308 Rogers Hall

Sourabh V. Apte. 308 Rogers Hall Sourabh V. Apte 308 Rogers Hall sva@engr.orst.edu 1 Topics Quick overview of Fluid properties, units Hydrostatic forces Conservation laws (mass, momentum, energy) Flow through pipes (friction loss, Moody

More information

Q1 Give answers to all of the following questions (5 marks each):

Q1 Give answers to all of the following questions (5 marks each): FLUID MECHANICS First Year Exam Solutions 03 Q Give answers to all of the following questions (5 marks each): (a) A cylinder of m in diameter is made with material of relative density 0.5. It is moored

More information

Chapter 5 Mass, Bernoulli, and Energy Equations Chapter 5 MASS, BERNOULLI, AND ENERGY EQUATIONS

Chapter 5 Mass, Bernoulli, and Energy Equations Chapter 5 MASS, BERNOULLI, AND ENERGY EQUATIONS Chapter 5 MASS, BERNOULLI, AND ENERGY EQUATIONS Conservation of Mass 5-C Mass, energy, momentum, and electric charge are conserved, and volume and entropy are not conserved during a process. 5-C Mass flow

More information

Isentropic Flow. Gas Dynamics

Isentropic Flow. Gas Dynamics Isentropic Flow Agenda Introduction Derivation Stagnation properties IF in a converging and converging-diverging nozzle Application Introduction Consider a gas in horizontal sealed cylinder with a piston

More information

cos(θ)sin(θ) Alternative Exercise Correct Correct θ = 0 skiladæmi 10 Part A Part B Part C Due: 11:59pm on Wednesday, November 11, 2015

cos(θ)sin(θ) Alternative Exercise Correct Correct θ = 0 skiladæmi 10 Part A Part B Part C Due: 11:59pm on Wednesday, November 11, 2015 skiladæmi 10 Due: 11:59pm on Wednesday, November 11, 015 You will receive no credit for items you complete after the assignment is due Grading Policy Alternative Exercise 1115 A bar with cross sectional

More information