MECHANICAL ENGINEERING FLUID ECHANICS · Manometers • Piezometers p gh A ... Application of...

MECHANICAL ENGINEERING –FLUID MECHANICS

MECHANICAL ENGINEERING – FLUIDMECHANICS

www.gradeup.co

2

Pressure (P):

• If F be the normal force acting on a surface of area A in contact with liquid,then pressure exerted by liquid on this surface is: P =F / A

• Units : N /m2 or Pascal (S.I.) and Dyne/cm2 (C.G.S.)

• Dimension

:

[P] =

[F]

[A] =

[MLT 2 ]

[L2 ]

=[ML-1T

-2 ]

• Atmospheric pressure:Its value on the surface of the earth at sea level is nearly

1.01*10 5N/ m2 or Pascal in S.I. other practical units of pressure are atmosphere, bar and torr (mm of Hg)

• 1atm = 1.01 *10 5Pa = 1.01bar

= 760 torr

• Fluid Pressure at aPoint:

Density ( ρ):

dp=dF

dA

• In a fluid, at a point, density ρ is defined as: Mass/volume

• In case of homogenous isotropic substance, it has no directional properties, so is a scalar.

• It has dimensions [ML-3] and S.I. unit kg/m3 while C.G.S. unit g/ccwith

1g / cc = 10 3kg / m 3

• Density of body = Density of substance

• Relative density or specific gravity which is defined as:

RD

Densit

Density

of body

of water

Specific Weight ( w): • It is defined as the weight per unit volume.

• Specific weight =Weight

= m.g

Volume Volume

Specific Gravity or Relative Density(s): • It is the ratio of specific weight of fluid to the specific weight of a standard fluid.

Standard fluid is water in case of liquid and H2 or air in case of gas.

Specific gravity=𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝐹𝑙𝑢𝑖𝑑

𝐷𝑒𝑛𝑠𝑖𝑡𝑦 𝑜𝑓 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐹𝑙𝑢𝑖𝑑 or

𝛾

𝛾𝑤 or

𝜌

𝜌𝑤

Where, 𝛾 Specific weight , and 𝜌

Density of water specific

http://www.gradeup.com/

MECHANICAL ENGINEERING – FLUIDMECHANICS www.gradeup.co

3

Specific Volume ( v): • Specific volume of liquid is defined as volume per unit mass. It is also defined as the

reciprocal of specific density.

• Specific volume =1/Density

Newton’s Law of viscosity

• 𝜏 = 𝜇𝑑𝑢

𝑑𝑦

Where 𝜏 = Shear Stress, 𝜇= Co-efficient of viscosity or absolute viscosity

𝑑𝑢

𝑑𝑦= Rate of angular deformation or rate of change of shear strain

Dynamic Viscosity and kinematic viscosity

• Dynamic Viscosity- Resistance offered by fluid to flow

• Units are Ns/m2 or Kg/ms

• 1 poise= 0.1 Ns/m2

• Kinematic Viscosity = 𝐷𝑦𝑛𝑎𝑚𝑖𝑐 𝑉𝑖𝑠𝑐𝑜𝑠𝑖𝑡𝑦(𝜇)

𝑀𝑎𝑠𝑠 𝐷𝑒𝑛𝑠𝑖𝑡𝑦 (𝜌)

• 1 Stoke= c.m2/s = 10-4 m2/s

Bulk Modulus

• Bulk modulus K= dP

−dVV

• Compressibility (β) = 1K = −dV

VdP= dρ

ρdP

Where K= Bulk modulus of elasticity, ρ= Density and V= Specific volume

Surface Tension

• The cohesive forces between liquid molecules are responsible for the phenomenon known as surface

tension

• Unit N/m

• Pressure inside drop P = 4σ

d

• Pressure inside bubble P = 8σ

d

• Pressure inside jet P = 2σ

d

Where d= Diameter of drop, P= Gauge pressure and σ= Surface tension



4

Capillary Action

• Height of water in capillary tube- 𝒉 =𝟒𝝈𝑪𝒐𝒔𝜽

𝝆𝒈𝒅

ℎ=rise in capillary 𝜎= Surface tension of water D= Diameter of tube

𝜃=Angle of contact between liquid and material

𝜃=<90o for water and glass and >90o for mercury and glass

Absolute Pressure

Hydrostatic law

𝒅𝑷

𝒅𝒉= 𝝆𝒈

𝐖𝐡𝐞𝐫𝐞 𝛒 = 𝐃𝐞𝐧𝐬𝐢𝐭𝐲 𝐚𝐧𝐝 𝐠 = 𝐀𝐜𝐜𝐞𝐥𝐞𝐫𝐚𝐭𝐢𝐨𝐧 𝐝𝐮𝐞 𝐭𝐨 𝐠𝐫𝐚𝐯𝐢𝐭𝐲

Manometers

• Piezometers

Ap gh=

abs atm gauge

abs atm vac

P P P

P P P

= +

= −



5

• U-Tube Manometer

1 1 2 2A m BP s gh s gh h gs P+ − − =

Where s1,s2 and sm are density of fluids in

manometer

Hydrostatic Forces on submerged bodies

• Vertical Planes

Total Force 𝐅 = 𝛒𝐠𝐀�̅�

• Center of Pressure

𝒉∗ =𝑰𝒈

𝑨�̅�+ �̅�

• Horizontal Surface

Total Force 𝐅 = 𝛒𝐠𝐀�̅�

• Center of Pressure

𝒉∗ = �̅�



www.gradeup.co

6

• Forces on Inclined surface .total C GF gh A=

• Centre of pressure

𝒉∗ =𝑰𝒈𝑺𝒊𝒏𝟐𝜽

𝑨𝒉𝑪.𝑮+ 𝒉𝑪.𝑮

• Curved Surface –

2 2

tan

R h v

v

h

F F F

F

F

= +

=

Vertical component of force Fv Weight of the liquid supported by the curved surface over it up to the free liquid surface

Horizontal component of force Fh Total pressure force on the vertical projected area of the curved surface

Completely submerged and floating at the interface of two liquids

1 1 2 2BF gV gV = + Where 1V and 2V are volumes

Body floating in a liquid

22BF V=



www.gradeup.co

7

Principal of floatation

• If the body weight is equal to the buoyant force, the body will float

BMg W F gV= = =

Condition of stability

• Fully submerged body

Stable Equilibrium: G below B Unstable Equilibrium: G above B Neutral Equilibrium: G coincides with B

• Floating body

Stable Equilibrium: M above G Unstable Equilibrium: M below G Neutral Equilibrium: M coincides with G

Metacentric height (GM)

• Metacentre radius (BM)

IBM

V=

• Metacentric Height (GM)

GM BM BG

IGM BG

V

= −

= −



www.gradeup.co

8

Where I= Second moment area about body surface vertical axis

V = Volume of water displaced

Time period of transverse oscillation of floating body

• 2 GKT

gGM= KG= Least radius of gyration, GM=Meta- Centric Height

Continuity Equation • ρ1A1V1 = ρ2A2V2

Where A,V are cross-section area of the flow and Velocity of flow respectively

For incompressible flow ρ=constant so

A1V1 = A2V2

Generalized differential Continuity Equation

• 𝑑𝜌

𝑑𝑡+

𝑑(𝜌𝑢)

𝑑𝑥+

𝑑(𝜌𝑣)

𝑑𝑦+

𝑑(𝜌𝑤)

𝑑𝑧= 0

Where u,v and w are the velocities in x,y,and z direction respectively

For steady incompressible two dimensional flow, ρ=constant and 𝑑𝜌

𝑑𝑡= 0

𝑑𝑢

𝑑𝑥+

𝑑𝑣

𝑑𝑦= 0

Velocity and Acceleration of Fluid Particle • Velo = ui⃗ + vj⃗ + wk⃗⃗

• Acceleration in X-direction 𝑎𝑥 =𝑑𝑢

𝑑𝑡= 𝑢

𝑑𝑢

𝑑𝑥+ 𝑣

𝑑𝑢

𝑑𝑦+ 𝑤

𝑑𝑢

𝑑𝑧+

𝑑𝑢

𝑑𝑡

Acceleration in Y-direction 𝑎𝑦 =𝑑𝑣

𝑑𝑡= 𝑢

𝑑𝑣

𝑑𝑥+ 𝑣

𝑑𝑣

𝑑𝑦+ 𝑤

𝑑𝑣

𝑑𝑧+

𝑑𝑣

𝑑𝑡

Acceleration in Y-direction 𝑎𝑧 =𝑑𝑤

𝑑𝑡= 𝑢

𝑑𝑤

𝑑𝑥+ 𝑣

𝑑𝑤

𝑑𝑦+ 𝑤

𝑑𝑤

𝑑𝑧+

𝑑𝑤

𝑑𝑡

For steady flow 𝑑𝑢

𝑑𝑡 =

𝑑𝑣

𝑑𝑡=

𝑑𝑤

𝑑𝑡= 0

𝑎 = √𝑎𝑥2 + 𝑎𝑦

2 + 𝑎𝑤2

Note- Local Acceleration due to increase in rate of velocity with respect to time at a point and

convective acceleration due to rate of change of position (𝑑𝑢

𝑑𝑡 =

𝑑𝑣

𝑑𝑡=

𝑑𝑤

𝑑𝑡= 0)



www.gradeup.co

9

Rotational fluid

ωz =1

2(

∂V

∂x−

∂U

∂y)

Vortex flow

• Free Vortex V*r=Constant

• Forced Vortex V=r ω and 𝐻 =𝜔2𝑟2

2𝑔

Velocity potential

• If ∅ is the velocity function, then

𝑢 = −𝜕∅

𝜕𝑥 , v=−

𝜕∅

𝜕𝑦 , 𝑤 = −

𝜕∅

𝜕𝑧

• Polar direction

𝑢𝑟 = −𝜕∅𝜕𝑟

,𝑢𝜃 = −1

𝑟

𝜕∅

𝜕𝜃

Stream Function

• If is the Stream function , then

𝑢 = −𝜕

𝜕𝑦 , v=

𝜕

𝜕𝑥

Equipotential line

• Condition for Equipotential line 𝑑∅ = 0

So 𝑑𝑦

𝑑𝑥= −

𝑢

𝑣

Line of constant stream function

• Condition for constant stream function 0d =

𝑑𝑦

𝑑𝑥=

𝑣

𝑢

Relation between stream function and velocity potential function

• ∂∅

∂x=

∂

∂y ,

∂∅

∂y= −

∂

∂x

Equation of motion

• 𝐹𝑥 = 𝐹𝑔 + 𝐹𝑃 + 𝐹𝛾 + 𝐹𝑡 + 𝐹𝐶

Where,

Gravity force 𝐹𝑔



www.gradeup.co

10

Pressure force 𝐹𝑃

Viscosity force 𝐹𝛾

Turbulance force 𝐹𝑡

Compressibility force 𝐹𝐶

• When Compressibility force 𝐹𝐶 is negligible

𝐹𝑥 = 𝐹𝑔 + 𝐹𝑃 + 𝐹𝛾 + 𝐹𝑡 is Reynold’s equation of motion

• 𝐹𝑡 is negligible 𝐹𝑥 = 𝐹𝑔 + 𝐹𝑃 + 𝐹𝛾 is Navier- Stokes equation of motion

• When flow is assumed to be ideal, 𝐹𝛾 = 0

𝐹𝑥 = 𝐹𝑔 + 𝐹𝑃 is Euler’s equation of motion

Euler’s equation of motion

• 𝑑𝑝

𝜌+ 𝑔𝑑𝑧 + 𝑣𝑑𝑣 = 0

Bernoulli’s equation

• 𝑃 + 𝜌𝑔𝑧 +𝜌𝑉2

2= 0

• P

ρg+ z +

V2

2g= 0

Where, P

ρg = Pressure head

Z = Potential head

V2

2g = Kinetic head

Application of Bernoulli’s theorem

• Venturimeter

Qact = Cd

a1a2√2gh

√a12 − a2

2

Where,

Cd=Co-efficient of venturimeter which is <1

a1, a2= Area of cross section, h= Head

• Orifice meter

Vena contracta 𝐶𝑐 =Area of vena contracta

Area of orific



www.gradeup.co

11

Discharge 𝐶𝑑 = 𝐶𝑐

√1−(aoa1

)2

√1−(aoa1

)2

𝑐𝑐2

where,

a1= area of cross section before vena contracta

ao=Area of orifice

• Pitot Tube

𝐕𝐚𝐜𝐭 = 𝐂𝐯√𝟐𝐠(𝐒𝐭𝐚𝐠𝐧𝐚𝐭𝐢𝐨𝐧 𝐡𝐞𝐚𝐝 − 𝐒𝐭𝐚𝐭𝐢𝐜 𝐡𝐞𝐚𝐝)

Where 𝐂𝐯 is co-efficient of velocity

Value of 𝒉𝒑 given by differential

• 𝒉𝒑 = 𝒚(𝑺𝒎

𝑺− 𝟏) where 𝑺𝒎 > 𝑺

• 𝒉𝒑 = 𝒚(𝟏 −𝑺𝒎

𝑺) where S > 𝑺𝒎

𝑺𝒎, 𝑺 are realtive density of manometric fluid and fluid flowing

Momentum Equation

• F.dt= d(mv) known as impulse- momentum equation

Force exerted on flowing fluid by a bend pipe

• If 𝜌1 = 𝜌2 then net force acting on a fluid

• 𝑃1𝐴1 − 𝑃1𝐴1𝐶𝑜𝑠𝜃 − 𝐹𝑥 =

Change in velocity in horizontal ∗ mass of fluid

• Fy = ρQ(change in velocity vertically) − P2A2Sinθ

• Resultant force = √Fx𝟐 + Fx

𝟐

Viscous flow • To be viscous flow Reynold number should be less than 2000

𝑅𝑒 =𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝑓𝑜𝑟𝑐𝑒

𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝑓𝑜𝑟𝑐𝑒=

𝜌𝑉𝐷

𝜇<2000

Where, v=velocity of flow, 𝜇= viscosity of flow, D= Diameter of piper and 𝜌= Density of fluid

• Flow of viscous fluid through circular pipe



www.gradeup.co

12

➢ Velocity u =1

4μ(−

∂P

∂x) [R2 − r2]

➢ Shear stress 𝜏 = (−∂P

∂x)

r

2

➢ Ratio of maximum to average velocity

Maximum Velocity

Average Velocity= 2

➢ Drop of pressure in given length

P1 − P2

ρg= hf =

32u̅μL

ρgD2

Also called Hagen Poiseuille Equation

Here u̅=average velocity, P1, P2 = Pressure at two different points in the pipe

• Flow of viscous fluid between two parallel plates

➢ Velocity u =1

2μ(−

∂P

∂x) [ty − y2]

➢ Shear Stress 𝜏 =1

2(−

∂P

∂x) [t − 2y]

➢ Ratio of maximum to average velocity

Maximum Velocity

Average Velocity=

3

2

➢ Drop of pressure in given length

P1−P2

ρg= hf =

12u̅μL

ρgt2

Kinetic energy correction factor

α =

K.E

secbased on actual velocity

K.E

secbased on average velocity

For laminar flow α=2 and for turbulent flow α=1.33

Momentum correction factor

β =

Momentum

secbased on actual velocity

Momentum

secbased on average velocity

For laminar flow β=1.33 and for turbulent flow β=1.20

Loss of head due to friction in viscous flow

hf =4flV2

2Dg

Where t is the thickness



www.gradeup.co

13

Where f= friction co-efficient

For laminar flow 𝑓 =16

𝑅𝑒 where Re is Reynold’s number

For turbulent flow, coefficient of friction f =0.079

Re14

Chezy’s Formula

V = C√mi , C = Chezy Constant = √ρg

f

i = Loss of head per unit length of pipe= ℎ𝑓

𝐿 (hydraulic slope tan θ)

m = Hydraulic mean depth

Mean velocity of flow m = Area (A)

Wetted Perimeter(P)

Minor losses in pipe

• Loss of head due to sudden enlargement

he =(V1 − V1)2

2g

• Loss of head due to sudden contraction

he =V2

2g(

1

Cc− 1)

2

or he = 0.5V2

2g when Cc=0.65

• Loss of head due to entrance

𝒉𝒆𝒏𝒕𝒓𝒂𝒏𝒄𝒆 = 0.5V2

2g

• Loss due to exit pipe

=𝑽𝟐

𝟐𝒈

• Loss due to obstruction

=V2

2g(

A

Cc(A−a)− 1)

2

• Losses due to bend

= K𝑽𝟐

𝟐𝒈 where k depends on bending of pipe

Hydraulic gradient and Total Energy line

• H.G.L=Pressure head+datum head

• T.G.L= Pressure head+datum head+Kinetic Head



www.gradeup.co

14

Flow through pipes in series or compound pipes

Major loss = Head loss due to friction in each pipe

While, minor loss = Entrance loss + Expansion loss +

Contraction loss + Exit loss

Flow will remain constant

Equivalent pipe in series pipe

𝐿

𝑑5=

𝐿1

𝑑15 +

𝐿2

𝑑25 +

𝐿3

𝑑35

Power Transmission through pipes

• P =ρgAV

1000(H − hf)

hf = loss due to friction • Efficency of power transmission

=𝑃𝑜𝑤𝑒𝑟 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑎𝑡 𝑜𝑢𝑡𝑙𝑒𝑡

𝑃𝑜𝑤𝑒𝑟 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑑 =

𝐻−ℎ𝑓

𝐻

• Condition for maximum transmission of power

𝐻 = 3ℎ𝑓 and = 66%

Dimensional analysis:

Quantity Symbol Dimensio

ns Mass m M

Length l L

Time t T

Temperature T θ

Velocity u LT -1



www.gradeup.co

15

Acceleration a LT -2

Momentum/Impulse mv MLT -1

Force F MLT -2

Energy - Work W ML 2T -2

Power P ML 2T -3

Moment of Force M ML 2T -2

Angular momentum - ML 2T -1

Angle η M 0L 0T 0

Angular Velocity ω T -1

Angular acceleration α T -2

Area A L 2

Volume V L 3

First Moment of Area Ar L 3

Second Moment of Area I L 4

Density ρ ML -3

Specific heat-

Constant

Pressure

C p L 2 T -2 θ -1

Elastic Modulus E ML -1T -2

Flexural Rigidity EI ML 3T -2

Shear Modulus G ML -1T -2

Torsional rigidity GJ ML 3T -2

Stiffness k MT -2

Angular stiffness T/η ML 2T -2

Flexibiity 1/k M -1T 2

Vorticity - T -1

Circulation - L 2T -1

Viscosity μ ML -1T -1

Kinematic Viscosity τ L 2T -1

Diffusivity - L 2T -1

Friction coefficient f /μ M 0L 0T 0

Restitution coefficient M 0L 0T 0

Specific heat-

Constant volume

C v L 2 T-2 θ -1



www.gradeup.co

16

Dimensionless number

• Reynold’s number (Re)

𝑅𝑒 =𝐼𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝐹𝑜𝑟𝑐𝑒

𝑉𝑖𝑠𝑐𝑜𝑢𝑠 𝐹𝑜𝑟𝑐𝑒 =

𝜌𝑉𝐷

𝜇

• Froude’s Number (Fe)

Fe = √Inertia Force

Gravity Force =

𝑉

√𝐿𝑔

• Euler’s Equation (Eu)

𝐸𝑢 = √𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝐹𝑜𝑟𝑐𝑒

𝑃𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝐹𝑜𝑟𝑐𝑒 =

𝑉

√𝑃𝜌⁄

• Weber’s number (We)

𝑊𝑒 =𝐼𝑛𝑒𝑟𝑡𝑖𝑎 𝐹𝑜𝑟𝑐𝑒

𝑆𝑢𝑟𝑓𝑎𝑐𝑒 𝐹𝑜𝑟𝑐𝑒 =

𝑉

√𝜎𝜌𝐿⁄

• Mach number (M)

𝑀 =𝐼𝑛𝑒𝑟𝑡𝑖𝑎

𝐸𝑙𝑎𝑠𝑡𝑖𝑐 𝐹𝑜𝑟𝑐𝑒=

𝑉

𝐶

Velocity distribution for turbulent flow in pipe

v = vmax + 2.5V∗loge (Y

R)

Where V∗=√𝜏𝑜

𝜌 Shear or friction velocity, Y=distance from pipe wall, 𝜌=Density

Displacement Thickness (𝛅∗)

𝛅∗ = ∫ {1 −𝑢

𝑈} 𝑑𝑦

𝛿

0

U=Stream Velocity, u=Velocity of fluid at the element, δ= boundary layer thickness

Momentum Thickness (θ)

θ = ∫𝑢

𝑈{1 −

𝑢

𝑈} 𝑑𝑦

𝛿

0

Energy Thickness (𝛅∗∗)

𝛅∗∗ = ∫𝑢

𝑈{1 −

𝑢2

𝑈2} 𝑑𝑦

𝛿

0



www.gradeup.co

17

Drag force on a flat plate due to boundary layer

τo

ρU2=

∂θ

∂x

Know as Von Karman momentum integral equation

Drag force on plate obtained by 𝐹𝐷 = ∫ ∆𝐹 = ∫ 𝜏𝑜𝐿

0𝑏𝑑𝑥 where, b if thickness of plate

Local (𝐂𝐃∗ )and average (𝐂𝐃) co-efficient of drag

CD∗ =

τ0

ρU2

2⁄

, CD =τ0

ρAU2

2⁄

Boundary condition for the velocity profile

• At y=0, u=0

• At y=𝛿, u=U 𝜕𝑢

𝜕𝑦= 0



www.gradeup.co

18

Saparation of boundary layer

Force exerted by jet on plate

• Force exerted by jet = mass of fluid striking per sec * change in velocity = ρa(V)*(V – 0) =ρa(V)2

Force exerted by jet on inclined plate

• Fn=ρa(V)2 Sinθ • Force component in X-direction = Fn Sinθ • Force component in Y-direction = Fn Cosθ

Force exerted by jet on curved plate

• Fx=ρaV(V+V cosθ) • Fy=ρaV(0-V sinθ)

Force exerted by jet on curved plate , moving with some velocity

• Fx=ρa(V-u)( (V-u)+ (V-u) cosθ) • Fy=ρa(V-u)(0-(V-u) sinθ)



www.gradeup.co

19

Forced exerted by a jet on a hinged plate

• Sinθ =ρaV2

w

Hydraulic Efficiency =𝐏𝐨𝐰𝐞𝐫 𝐝𝐞𝐥𝐢𝐯𝐞𝐫𝐞𝐝 𝐭𝐨 𝐫𝐮𝐧𝐧𝐞𝐫

𝐏𝐨𝐰𝐞𝐫 𝐬𝐮𝐩𝐩𝐥𝐢𝐞𝐝 𝐭𝐨 𝐢𝐧𝐥𝐞𝐭=

𝐑.𝐏

𝐖.𝐏

Mechanical Efficiency =𝐏𝐨𝐰𝐞𝐫 𝐚𝐭 𝐭𝐡𝐞 𝐬𝐡𝐚𝐟𝐭 𝐨𝐟 𝐭𝐡𝐞 𝐭𝐮𝐫𝐛𝐢𝐧𝐞

𝐏𝐨𝐰𝐞𝐫 𝐝𝐞𝐥𝐢𝐯𝐞𝐫𝐞𝐝 𝐛𝐲 𝐰𝐚𝐭𝐞𝐫 𝐭𝐨 𝐭𝐡𝐞 𝐫𝐮𝐧𝐧𝐞𝐫=

𝐒.𝐏

𝐑.𝐏

Volumetric Efficiency = 𝐕𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 𝐰𝐚𝐭𝐞𝐫 𝐚𝐜𝐭𝐮𝐚𝐥𝐥𝐲 𝐬𝐭𝐫𝐢𝐤𝐢𝐧𝐠 𝐭𝐡𝐞 𝐫𝐮𝐧𝐧𝐞𝐫

𝐕𝐨𝐥𝐮𝐦𝐞 𝐨𝐟 𝐰𝐚𝐭𝐞𝐫 𝐬𝐮𝐩𝐩𝐥𝐢𝐞𝐝 𝐭𝐨 𝐭𝐡𝐞 𝐭𝐮𝐫𝐛𝐢𝐧𝐞

Overall Efficiency = 𝐒𝐡𝐚𝐟𝐭 𝐏𝐨𝐰𝐞𝐫

𝐖𝐚𝐭𝐞𝐫 𝐏𝐨𝐰𝐞𝐫= Hydraulic Efficiency * Mechanical Efficiency



www.gradeup.co

20

Work done by Pelton turbine

• Work 𝑊 = 𝜌𝑎𝑉1[𝑉𝜔1 − 𝑉𝜔2]𝑢 where 𝑉𝜔1, 𝑉𝜔2are whirl velocity

Hydraulic Efficiency

• Hydraulic Efficiency =𝟐[𝐕𝛚𝟏±𝐕𝛚𝟐]𝐮

𝐕𝟏𝟐

• When blade velocity= (inlet velocity of jet)/2 then,

Maximum efficiency 𝟏+𝑪𝒐𝒔∅

𝟐

Degree of reaction

R =Change of pressure energy inside the runner

Change in total energy

Specific Speed

• For turbine 𝑁𝑠 =𝑁√𝑃

𝐻5

4⁄ where P= power, H= head and N= number of rotation

• Dimensionless Specific speed 𝑁𝑠 =𝑁√𝑃

(𝑔𝐻)5

4⁄

Turbine Specific Speed (S.I)

Specific Speed (M.K.S)

Pelton 8.5 to 30 10 to 35

Pelton with two jets 30 to 51 35 to 60

Francis 51 to 255 60 to 300

Kaplan and propeller 255 to 860 300 to 1000

For Pumps, 𝑁𝑠 =𝑁√𝑄

𝐻3

4⁄ where Q is discharge

Unit quantities

• Unit speed (Nu): N = Nu√H

• Unit Power (Pu): P = Pu ∗ H3

2⁄

• Unit discharge (Qu): Q = Qu√H

Model laws of turbine

• 𝑄

𝑁𝐷3 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

• 𝑄

√𝐻𝐷2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡



www.gradeup.co

21

• 𝐻

𝑁2𝐷2 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

• 𝑃

𝑁3𝐷5 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

• 𝑃

𝐻3

2⁄ 𝐷2= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Net Positive suction Head in pump

• NPSH= Pressure head + Static head - Vapor pressure head of your product – Friction head loss



www.gradeup.co

22

=


MECHANICAL ENGINEERING FLUID ECHANICS · Manometers • Piezometers p gh A ... Application of...

Documents

Transcript of MECHANICAL ENGINEERING FLUID ECHANICS · Manometers • Piezometers p gh A ... Application of...