ME 33, Fluid Flow Chapter 5: Mass, Bernoulli, and Energy ...

ME 33, FluidFlow

Chapter 5:Mass,

Bernoulli, andEnergy

Equations

Introduction

Conservationof Mass

MechanicalEnergy andEfficiency

GeneralEnergyEquation

EnergyAnalysis ofSteady Flows

The BernoulliEquation

ME 33, Fluid FlowChapter 5: Mass, Bernoulli, and Energy

Equations

Eric G. Paterson

Department of Mechanical and Nuclear EngineeringThe Pennsylvania State University

Spring 2005

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





Introduction

This chapter deals with 3 equations commonly used influid mechanics

The mass equation is an expression of the conservationof mass principle.The Bernoulli equation is concerned with theconservation of kinetic, potential, and flow energies of afluid stream and their conversion to each other.The energy equation is a statement of the conservationof energy principle.

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





Objectives

After completing this chapter, you should be able toApply the mass equation to balance the incoming andoutgoing flow rates in a flow system.Recognize various forms of mechanical energy, andwork with energy conversion efficiencies.Understand the use and limitations of the Bernoulliequation, and apply it to solve a variety of fluid flowproblems.Work with the energy equation expressed in terms ofheads, and use it to determine turbine power output andpumping power requirements.

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





Conservation of Mass

Conservation of mass principle is one of the mostfundamental principles in nature.

Mass, like energy, is a conserved property, and itcannot be created or destroyed during a process.

For closed systems mass conservation is implicit sincethe mass of the system remains constant during aprocess.

For control volumes, mass can cross the boundarieswhich means that we must keep track of the amount ofmass entering and leaving the control volume.

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





Mass and Volume Flow Rates

The amount of mass flowingthrough a control surface per unittime is called the mass flow rateand is denoted m.

The dot over a symbol is used toindicate time rate of change.

Flow rate across the entirecross-sectional area of a pipe orduct is obtained by integration

m =

∫Ac

δm =

∫Ac

ρVn dAc

While this expression for m isexact, it is not always convenientfor engineering analyses.

ME 33, FluidFlow

Chapter 5:Mass,


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Introduction

Conservationof Mass





Average Velocity and Volume Flow Rate

Integral in m can be replaced withaverage values of ρ and Vn

Vavg =1

Ac

∫Ac

Vn dAc

For many flows variation of ρ isvery small: m = ρVavgAc .

Volume flow rate V is given by

V =

∫Ac

Vn dAc = VavgAc = VAc

Note: many textbooks use Qinstead of V for volume flow rate.

Mass and volume flow rates arerelated by m = ρV

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





Conservation of Mass Principle

The conservation of massprinciple can be expressed asmin − mout = dmCV

dt

Where min and mout are the totalrates of mass flow into and out ofthe CV, and dmCV /dt is the rate ofchange of mass within the CV.

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





Conservation of Mass Principle

For CV of arbitrary shape,1 rate of change of mass within the

CVdmCV

dt=

ddt

∫CV

ρ dV

2 net mass flow rate‘ mnet =

∫CS

δm =∫CS

ρVn dA =

∫CS

ρ(

~V · ~n)

dA

Therefore, general conservation ofmass for a fixed CV is:ddt

∫CV

ρ dV +

∫CS

ρ(

~V · ~n)

dA = 0

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





Steady–Flow Processes

For steady flow, the total amountof mass contained in CV isconstant.

Total amount of mass enteringmust be equal to total amount ofmass leaving.∑

in

m =∑out

m

For incompressible flows,∑in

V =∑out

V =⇒∑in

VnAn =∑out

VmAm

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





Mechanical Energy

Mechanical energy can be defined as the form ofenergy that can be converted to mechanical workcompletely and directly by an ideal mechanical devicesuch as an ideal turbine.

Flow P/ρ, kinetic V 2/g, and potential gz energy are theforms of mechanical energy emech = P

ρ + V 2

2 + gz

Mechanical energy change of a fluid duringincompressible flow becomes

∆emech = P2−P1ρ +

V 22−V 2

12 + g(z2 − z1)

In the absence of loses, ∆emech represents the worksupplied to the fluid (∆emech > 0) or extracted from thefluid (∆emech < 0).

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





Efficiency

Transfer of emech is usually accomplished by a rotatingshaft: shaft work

Pump, fan, propulsor: receives shaft work (e.g., from anelectric motor) and transfers it to the fluid asmechanical energy

Turbine: converts emech of a fluid to shaft work.

In the absence of irreversibilities (e.g., friction),mechanical efficiency of a device or process can bedefined as ηmech =

Emech,outEmech,in

= 1 − Emech,lossEmech,in

If ηmech < 100%, losses have occurred duringconversion.

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





Pump and Turbine Efficiencies

In fluid systems, we are usuallyinterested in increasing thepressure, velocity, and/or elevationof a fluid.

In these cases, efficiency is betterdefined as the ratio of (supplied orextracted work) vs. rate of increasein mechanical energy ∆Emech,fluid .

ηpump =∆Emech,fluid

Wshaft,in

ηturbine =Wshaft,out

|∆Emech,fluid |Overall efficiency must includemotor or generator efficiency.

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





General Energy Equation

One of the most fundamental laws in nature is the 1stlaw of thermodynamics , which is also known as theconservation of energy principle .

It state that energy can be neither created nordestroyed during a process; it can only change forms

Falling rock, picks up speed as PEis converted to KE.

If air resistance is neglected, PE +KE = constant.

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass






The energy content of a closedsystem can be changed by twomechanisms: heat transfer Q andwork transfer W.

Conservation of energy for aclosed system can be expressedin rate form as

Qnet ,in + Wnet ,in =dEsys

dtNet rate of heat transfer to thesystem: Qnet ,in = Qin − Qout

Net power input to the system:Wnet ,in = Win − Wout

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass






Recall general RTTdBsys

dt=

ddt

∫CV

ρb dV +

∫CS

ρb(

~Vr · ~n)

dA

“Derive” energy equation using B = E and b = edEsys

dt= Qnet ,in + Wnet ,in =

ddt

∫CV

ρe dV +

∫CS

ρe(

~Vr · ~n)

dA

Break power W into rate of shaft and pressure workWnet ,in = Wshaft ,net ,in + Wpressure,net ,in =

Wshaft ,net ,in −∫

AP

(~V · ~n

)dA

ME 33, FluidFlow

Chapter 5:Mass,


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Introduction

Conservationof Mass






Where does expression for pressurework come from?

When piston moves down ds under theinfluence of F = PA, the work done onthe system is δWboundary = PA ds.

If we divide both sides by dt , we haveδWpressure = δWboundary = PAds

dt =PAVpiston

For generalized control volumes:

δWpressure = −P dA Vn = −P dA(

~V · ~n)

Note sign conventions:~n is outward pointing normalNegative sign ensures that work doneis positive when is done on the system.

ME 33, FluidFlow

Chapter 5:Mass,


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Introduction

Conservationof Mass






Moving integral for rate of pressure work to RHS ofenergy equation results in:

Qnet ,in + Wshaft ,net ,in =ddt

∫CV

ρe dV +

∫CS

(Pρ

+ e)

ρ(

~Vr · ~n)

dA

Recall that P/ρ is the flow work , which is the workassociated with pushing a fluid into or out of a CV perunit mass.

ME 33, FluidFlow

Chapter 5:Mass,


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Introduction

Conservationof Mass






As with the mass equation, practical analysis is oftenfacilitated as averages across inlets and exits

Qnet ,in + Wshaft ,net ,in =ddt

∫CV

ρe dV +∑out

m(

Pρ

+ e)−

∑in

m(

Pρ

+ e)

Note that m =∫

ACρ

(~V · ~n

)dAc

Since e = u + ke + pe = u + V 2

2 + gz

Qnet ,in + Wshaft ,net ,in =

ddt

∫CV

ρe dV +∑out

m(

Pρ

+ u +V 2

2+ gz

)−

∑in

m(

Pρ

+ u +V 2

2+ gz

)

ME 33, FluidFlow

Chapter 5:Mass,


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Introduction

Conservationof Mass





Energy Analysis of Steady Flows

Qnet ,in + Wshaft ,net ,in =∑out

m(

h +V 2

2+ gz

)−

∑in

m(

h +V 2

2+ gz

)

For steady flow, time rate of change of the energycontent of the CV is zero.

This equation states: the net rate of energy transfer to aCV by heat and work transfers during steady flow isequal to the difference between the rates of outgoingand incoming energy flows with mass.

ME 33, FluidFlow

Chapter 5:Mass,


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Introduction

Conservationof Mass






For single-stream devices , m isconstant.

qnet ,in + wshaft ,net ,in =

h2 − h1 +V 2

2−V 21

2 + g(z2 − z1)

wshaft ,net ,in + P1ρ1

+V 2

12 + gz1 =

P2ρ2

+V 2

22 + gz2 +

(u2 − u1 − qnet ,in

)P1ρ1

+V 2

12 + gz1 + wpump =

P2ρ2

+V 2

22 + gz2 + wturbine + emech,loss

ME 33, FluidFlow

Chapter 5:Mass,


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Introduction

Conservationof Mass






Divide by g to get each term in units of length

P1ρ1g +

V 21

2g + z1 + hpump = P2ρ2g +

V 22

2g + z2 + hturbine + hL

Magnitude of each term is now expressed as an equivalentcolumn height of fluid, i.e., head.

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass





The Bernoulli Equation

If we neglect piping losses, and have a system withoutpumps or turbines,

P1

ρ1g+

V 21

2g+ z1 =

P2

ρ2g+

V 22

2g+ z2

orPρg

+V 2

2g+ z = C1 P + ρ

V 2

2+ ρgz = C2

This is the Bernoulli equation

It can also be derived using Newton’s second law ofmotion (see text, p. 187).

3 terms correspond to: Static, dynamic, and hydrostatichead (or pressure).

ME 33, FluidFlow

Chapter 5:Mass,


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Introduction

Conservationof Mass





HGL and EGL

It is often convenient to plotmechanical energy graphicallyusing heights.

Hydraulic Grade LineHGL = P

ρg + z

Energy Grade Line (or totalenergy) EGL = P

ρg + V 2

2g + z

ME 33, FluidFlow

Chapter 5:Mass,


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Introduction

Conservationof Mass






The Bernoulli equation is anapproximate relation betweenpressure, velocity, and elevationand is valid in regions of steady,incompressible flow where netfrictional forces are negligible.

Equation is useful in flow regionsoutside of boundary layers andwakes.

ME 33, FluidFlow

Chapter 5:Mass,


Equations

Introduction

Conservationof Mass






Limitations on the use of the Bernoulli Equation

Steady flow: d/dt = 0

Frictionless flow

No shaft work: wpump = wturbine = 0

Incompressible flow: ρ = constant

No heat transfer: qnet ,in = 0

Applied along a streamline (except for irrotational flow).

ME 33, Fluid Flow Chapter 5: Mass, Bernoulli, and Energy ...

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