Lecture 6 Momentum Analysis of Flow Systems

RTT, Mass, Bernoulli, and Energy Equations: 

RTT, Mass, Bernoulli, and Energy Equations Chapter 5

Introduction: 

Introduction Reynolds Transport Theorem (RTT) provides a link between the system approach and the control volume approach Three equations which are commonly used in fluid mechanics The mass equation is an expression of the conservation of mass principle. The Bernoulli equation is concerned with the conservation of kinetic, potential, and flow energies of a fluid stream and their conversion to each other. The energy equation is a statement of the conservation of energy principle.

Objectives: 

Objectives After completing this chapter, you should be able to Understand the usefulness of the Reynolds Transport Theorem Apply the mass equation to balance the incoming and outgoing flow rates in a flow system. Understand the use and limitations of the Bernoulli equation, and apply it to solve a variety of fluid flow problems. Work with the energy equation expressed in terms of heads, and use it to determine turbine power output and pumping power requirements.

Reynolds—Transport Theorem (RTT): 

Reynolds—Transport Theorem (RTT) A system is a quantity of matter of fixed identity. No mass can cross a system boundary. A control volume is a region in space chosen for study. Mass can cross a control surface. CV fixed, nondeformable System deformable

Slide 5: 

28 August 2011 5 Fixed Control Volume for nozzle – stress analysis Control volume moving at ship speed for drag – force analysis Control volume deforming with cylinder for transient pressure – variation analysis

Reynolds—Transport Theorem (RTT): 

Reynolds—Transport Theorem (RTT) The fundamental conservation laws (conservation of mass, energy, and momentum) apply directly to systems. However, in most fluid mechanics problems, control volume analysis is preferred over system analysis (for the same reason that the Eulerian description is usually preferred over the Lagrangian description). Therefore, we need to transform the conservation laws from a system to a control volume. This is accomplished with the Reynolds transport theorem (RTT).

Reynolds—Transport Theorem (RTT): 

Reynolds—Transport Theorem (RTT)

Reynolds—Transport Theorem (RTT): 

Reynolds—Transport Theorem (RTT) the time rate of change of the property B of the system is equal to the time rate of change of B of the control volume plus the net flux of B out of the control volume by mass crossing the control surface.

Reynolds—Transport Theorem (RTT): 

Reynolds—Transport Theorem (RTT) The total amount of property B within the control volume must be determined by integration: Therefore, the system-to-control- volume transformation for a fixed control volume:

Reynolds—Transport Theorem (RTT): 

Reynolds—Transport Theorem (RTT) Material derivative (differential analysis): General RTT, nonfixed CV (integral analysis): Mass Momentum Energy Angular momentum B, Extensive properties m E b, Intensive properties 1 e we can apply RTT to conservation of mass, energy, linear momentum, and angular momentum.

Reynolds—Transport Theorem (RTT): 

Reynolds—Transport Theorem (RTT) Interpretation of the RTT: Time rate of change of the property B of the system is equal to (Term 1) + (Term 2) Term 1: the time rate of change of B of the control volume Term 2: the net flux of B out of the control volume by mass crossing the control surface

RTT Special Cases: 

RTT Special Cases For moving and/or deforming control volumes, Where the absolute velocity V in the second term is replaced by the relative velocity V r = V -V CS V r is the fluid velocity expressed relative to a coordinate system moving with the control volume.

RTT Special Cases: 

RTT Special Cases For steady flow, the time derivative drops out, = 0 For control volumes with well-defined inlets and outlets

Choosing a Control Volume: 

Choosing a Control Volume CV is arbitrarily chosen by fluid dynamicist, however, selection of CV can either simplify or complicate analysis. Clearly define all boundaries. Analysis is often simplified if CS is normal to flow direction. Clearly identify all fluxes crossing the CS. Clearly identify forces and torques of interest acting on the CV and CS. Fixed, moving, and deforming control volumes . For moving CV, use relative velocity, For deforming CV, use relative velocity all deforming control surfaces,

Reynolds—Transport Theorem (RTT): 

Reynolds—Transport Theorem (RTT) There is a direct analogy between the transformation from Lagrangian to Eulerian descriptions (for differential analysis using infinitesimally small fluid elements) and the transformation from systems to control volumes (for integral analysis using large, finite flow fields).

Remarks about RTT: 

The RTT is useful for transforming conservation equations from their naturally occurring systems forms to their control volume. The RTT can be applied to any control volume, fixed, moving, or deforming. The RTT has an unsteady term and can be applied to unsteady problems. The extensive property B (or its intensive form b) in the RTT can be any property of the fluid – scalar, vector, or even tensor. Remarks about RTT

Conservation of Mass: 

Conservation of Mass Conservation of mass principle is one of the most fundamental principles in nature. Mass, like energy, is a conserved property, and it cannot be created or destroyed during a process. For closed systems mass conservation is implicit since the mass of the system remains constant during a process. For control volumes, mass can cross the boundaries which means that we must keep track of the amount of mass entering and leaving the control volume.

Mass and Volume Flow Rates: 

Mass and Volume Flow Rates The amount of mass flowing through a control surface per unit time is called the mass flow rate and is denoted The dot over a symbol is used to indicate time rate of change. Flow rate across the entire cross-sectional area of a pipe or duct is obtained by integration While this expression for is exact, it is not always convenient for engineering analyses.

Average Velocity and Volume Flow Rate: 

Average Velocity and Volume Flow Rate Integral in can be replaced with average values of  and V n For many flows, variation of  is very small: Volume flow rate is given by Note: many textbooks use Q instead of for volume flow rate. Mass and volume flow rates are related by

Conservation of Mass Principle: 

Conservation of Mass Principle The conservation of mass principle can be expressed as Where and are the total rates of mass flow into and out of the CV, and dm CV /dt is the rate of change of mass within the CV.

Conservation of Mass Principle: 

Conservation of Mass Principle For CV of arbitrary shape, rate of change of mass within the CV net mass flow rate Therefore, general conservation of mass for a fixed CV is:

Steady—Flow Processes: 

Steady—Flow Processes For steady flow, the total amount of mass contained in CV is constant. Total amount of mass entering must be equal to total amount of mass leaving For incompressible flows,

Mechanical Energy: 

Mechanical Energy Mechanical energy can be defined as the form of energy that can be converted to mechanical work completely by an ideal mechanical device. Flow P/  , kinetic V 2 /g, and potential gz energy are the forms of mechanical energy e mech = P/  + V 2 /g + gz Mechanical energy change of a fluid during incompressible flow becomes In the absence of loses,  e mech represents the work supplied to the fluid (  e mech >0) or extracted from the fluid (  e mech <0).

Efficiency: 

Efficiency Transfer of e mech is usually accomplished by a rotating shaft: shaft work Pump, fan, propulsion: receives shaft work (e.g., from an electric motor) and transfers it to the fluid as mechanical energy Turbine: converts e mech of a fluid to shaft work. In the absence of irreversibilities (e.g., friction), mechanical efficiency of a device or process can be defined as If  mech < 100%, losses have occurred during conversion.

Pump and Turbine Efficiencies: 

Pump and Turbine Efficiencies In fluid systems, we are usually interested in increasing the pressure, velocity, and/or elevation of a fluid. In these cases, efficiency is better defined as the ratio of supplied or extracted work vs. rate of increase in mechanical energy Overall efficiency must include motor or generator efficiency.

The Bernoulli Equation: 

The Bernoulli Equation The Bernoulli equation is an approximate relation between pressure, velocity, and elevation and is valid in regions of steady, incompressible flow where net frictional forces are negligible . Equation is useful in flow regions outside of boundary layers and wakes.

The Bernoulli Equation: 

The Bernoulli Equation If we neglect piping losses, and have a system without pumps or turbines This is the Bernoulli equation It can also be derived using Newton's second law of motion . 3 terms correspond to: Static, dynamic, and hydrostatic head (or pressure).

Force balance across streamlines: 

Force balance across streamlines Steady Incompressible flow Unsteady compressible flow

Static, Dynamic and Stagnation Pressures: 

Static, Dynamic and Stagnation Pressures

Slide 39: 

P is the static pressure (it does not incorporate any dynamic effects); it represents the actual pressure of the fluid. This is the same as the pressure used in thermodynamics and property tables. V 2 /2 is the dynamic pressure ; it represents the pressure rise when the fluid in motion is brought to a stop isentropically.  gz is the hydrostatic pressure , which is not pressure in a real sense since its value depends on the reference level selected; it accounts for the elevation effects, i.e., of fluid weight on pressure. The sum of the static, dynamic, and hydrostatic pressures is called the total pressure. Therefore, the Bernoulli equation states that the total pressure along a streamline is constant.

Slide 40: 

When a stationary body is immersed in a flowing stream, the fluid is brought to a stop at the nose of the body (the stagnation point). The flow streamline that extends from far upstream to the stagnation point is called the stagnation streamline The sum of the static and dynamic pressures is called the stagnation pressure, and it is expressed as

Slide 41: 

A static pressure tap is simply a small hole drilled into a wall such that the plane of the hole is parallel to the flow direction. It measures the static pressure. A Pitot tube is a small tube with its open end aligned into the flow so as to sense the full impact pressure of the flowing fluid. It measures the stagnation pressure. In situations in which the static and stagnation pressure of a flowing liquid are greater than atmospheric pressure, a vertical transparent tube called a piezometer tube (or simply a piezometer) can be attached to the pressure tap and to the Pitot tube, as sketched. The liquid rises in the piezometer tube to a column height (head) that is proportional to the pressure being measured.

The Bernoulli Equation: 

The Bernoulli Equation Limitations on the use of the Bernoulli Equation Steady flow: d/dt = 0 Frictionless flow No shaft work: w pump =w turbine = 0 Incompressible flow:  = constant No heat transfer: q net,in =0 Applied along a streamline (except for irrotational flow)

HGL and EGL: 

HGL and EGL It is often convenient to plot mechanical energy graphically using heights to facilitate visualization of the various terms of the Bernoulli equation. Hydraulic Grade Line Energy Grade Line (or total energy)

Following points about EGL & HGL are worth noting: 

For Stationary bodies such as reservoir or lake , EGL and HGL coincide with the free surface of the liquid (z= EGL=HGL), since the velocity is zero and static (gage) pressure is zero. For open channel flow , the HGL coincides with the free surface of the liquid, and the EGL is a distance V 2 /2g above the free surface. At a pipe exit , the pressure head is zero (atmospheric pressure) and thus the HGL coincides with the pipe exit. The mechanical energy loss due to frictional effects (conversion to thermal energy) causes the EGL and HGL to slope downward in the direction of flow. The slope is a measure of the pipe loss . Following points about EGL & HGL are worth noting

Slide 46: 

• A steep jump occurs in EGL and HGL whenever mechanical energy is added to the fluid (by a pump, for example). Likewise, a steep drop occurs in EGL and HGL whenever mechanical energy is removed from the fluid (by a turbine, for example)

Slide 47: 

The pressure (gage) of a fluid is zero at locations where the HGL intersects the fluid. The pressure in a flow section that lies above the HGL is negative, and the pressure in a section that lies below the HGL is positive. Therefore, an accurate drawing of a piping system and the HGL can be used to determine the regions where the pressure in the pipe is negative (below the atmospheric pressure).

Applications Of The Bernoulli Equation: 

Applications Of The Bernoulli Equation Water Discharge from a Large Tank Spraying Water into the Air

Slide 49: 

Velocity Measurement by a Pitot Tube Siphoning out Gasoline from a Fuel Tank

General Energy Equation: 

General Energy Equation One of the most fundamental laws in nature is the 1 st law of thermodynamics, which is also known as the conservation of energy principle. It states that energy can be neither created nor destroyed during a process; it can only change forms Falling rock, picks up speed as PE is converted to KE. If air resistance is neglected, PE + KE = constant

General Energy Equation: 

General Energy Equation The energy content of a closed system can be changed by two mechanisms: heat transfer Q and work transfer W. Conservation of energy for a closed system can be expressed in rate form as Net rate of heat transfer to the system: Net power input to the system:

General Energy Equation: 

General Energy Equation When piston moves down ds under the influence of F=PA, the work done on the system is  W boundary =PAds. If we divide both sides by dt, we have For generalized control volumes: Note sign conventions: is outward pointing normal Negative sign ensures that work done is on the system, positive when work is done by the system.

General Energy Equation: 

General Energy Equation Recall general RTT “Derive” energy equation using B=E and b=e Break power into rate of shaft and pressure work

General Energy Equation: 

General Energy Equation Moving integral for rate of pressure work to RHS of energy equation results in: Recall that P/  is the flow work, which is the work associated with pushing a fluid into or out of a CV per unit mass.

General Energy Equation: 

General Energy Equation As with the mass equation, practical analysis is often facilitated as averages across inlets and exits Since e=u+ke+pe = u+V 2 /2+gz

Energy Analysis of Steady Flows: 

Energy Analysis of Steady Flows For steady flow, time rate of change of the energy content of the CV is zero. This equation states: the net rate of energy transfer to a CV by heat and work transfers during steady flow is equal to the difference between the rates of outgoing and incoming energy flows with mass.

Energy Analysis of Steady Flows: 

Energy Analysis of Steady Flows For single-stream devices, mass flow rate is constant.

Slide 58: 

28 August 2011 58

Energy Analysis of Steady Flows: 

Energy Analysis of Steady Flows Divide by g to get each term in units of length Magnitude of each term is now expressed as an equivalent column height of fluid, i.e., Head

Kinetic Energy Correction Factor, : 

Kinetic Energy Correction Factor,  Gaspard Coriolis showed the kinetic energy of the fluid stream obtained by V 2 /2 is not the same as the actual kinetic energy of the fluid film since the square of the sum is not equal to the sum of squares of its components. This error can be corrected by replacing the V 2 /2 by  V 2 avg /2 Where  Is the kinetic energy correction factor

=2 for fully developed laminar pipe flow  = 1.04 to 1.11 for a fully developed turbulent flow in a round pipe: 

=2 for fully developed laminar pipe flow  = 1.04 to 1.11 for a fully developed turbulent flow in a round pipe When the kinetic energy correction factor are included, the energy equation for steady incompressible flow become where

Slide 62: 

The kinetic energy correction factors are usually disregarded in an elementary analysis since (1) most flows encountered in practice are turbulent, for which the correction factor is near unity. (2) the kinetic energy terms are usually small relative to the other terms in the energy equation, and multiplying them by a factor less than 2.0 does not make much difference. Besides, when the velocity and thus the kinetic energy are high, the flow turns turbulent. Therefore, we will not consider the kinetic energy correction factor in the analysis.