ch12_momentum and mechanical energy equations

8/19/2019 Ch12_Momentum and Mechanical Energy Equations

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Thermo-fluid Engineering (MEC 2920) 1

Fluid Mechanics

Chapter 12: The Momentum andMechanical

Energy Equations


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Viscosity

• For Newtonian fluids ;

is called the viscosity

• expresses its resistanceto shearing flows


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Newtonian vs Non-Newtonian

• A fluid that behaves according toNewton's law, with aviscosity μ that is independent ofthe stress, is said tobe Newtonian. !ases,water and "any co""on

li#uids can be consideredNewtonian in ordinary conditionsand contexts.

• $here are "any non%Newtonianfluids that significantly deviatefro" that law in so"e way orother. &or exa"ple (hear thic)ening li#uids, whose

viscosity increases with the rate ofshear stress.

(hear thinning li#uids, whose

viscosity decreases with the rate ofshear stress. *ingha" plastics that behave as a

solid at low stresses but flows as aviscous fluid at high stresses


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Thermo-fluid Engineering (MEC 2920) +

Viscosity

ine"atic viscosity

• Inviscid flow we neglect viscosity effects i.e. µ -

• Incompressible Flow density is considered constant

• Steady flow flow properties /density, pressure, te"perature,velocity 0 do not change with ti"e

• alues of viscosity for several co""on gases and li#uids are listed in

the tables in Appendix FM-1 .


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Thermo-fluid Engineering (MEC 2920)

Momentum !uation

For solids

Newton s second law of "otion for solid particle

F = m a - d(mV)/ dt ,

F is the resultant force acting on the particlea is the accelerationmV is linear momentum

the resultant force on the particle is e#ual to the ti"e rate of change ofthe particle s "o"entu"


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Momentum !uation

For fluids" Newton#s second low


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Momentum !uation

At steady state , the total a"ount of "o"entu" contained in the

control volu"e is constant with ti"e.

6o"entu" e#uation


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Streamlines

(trea"lines are lines that are tangent to the velocity vector at any location inthe flow

&or steady flow a strea"line can be thought of as the path along which a fluidparticle "oves when traveling fro" one location in the flow, point /10, to anotherlocation, point /20.


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%he $ernoulli !uation

9hen shear forces due to viscosity /friction0 are negligible, Newton ssecond law leads to /for inco"pressible flow0

p (tatic pressure /ther"odyna"ic pressure0γ , hydrostatic pressure: ρ 2 dynamic pressure

p! : ρ 2 "tagnation pressure p! : ρ 2! γ total pressure


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*ernoulli e#uation between points /10 and /20

p1 static pressure ; p 2 stagnation pressure


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Free 'et

Vertical %an(


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xample

• Water is fowing rom a garden hose. child places histhum! to co"er most o the hose outlet# causing a thin

$et high speed water to emerge. The pressure in the hose $ust upstream o his thum! is %&&'(a. ) the hose is heldupward# what is the ma*imum height that the $et couldachie"e+


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Thermo-fluid Engineering (MEC 2920) 2+

xample

p & ' γ ( !h) = p % ' γ ' " γ h


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– When )t is pplied • In situations where the Bernoulli equation can not be applied, i.e. when

the fow is viscous and/or there is mechanical device such as a turbine, ora pump within the fow passage.

– ,ead loss - h L• Accounts or the irreversible conversion o mechanical energ internal

energ due to riction.

Mechanical ner)y !uation

pu"p adds head /or "echanical energy0 to what was available at the inlet, whereas both aturbine and friction reduce the a"ount of head /or "echanical energy0 available at the outlet.


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Momentum and Mechanical ner)y !uations


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Momentum and Mechanical ner)y !uations


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Thermo-fluid Engineering (MEC 2920) 3+3+

Assi)n

*hapter 1+ " 1 , 1. /0 /

ch12_momentum and mechanical energy equations

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