One Dimensional Flow of Blissful Fluid -III P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Always Start with simplest Inventions……..

One Dimensional Flow of Blissful Fluid -III

P M V Subbarao Professor

Mechanical Engineering DepartmentI I T Delhi

Always Start with simplest Inventions……..

Differential Form of Momentum Equation

dpududx

dp

dx

duu 0

One dimensional steady inviscid flow :

dpudu

udp

du

1

The relation between pressure and velocity is continuous.

Differential Form of Energy Equation

One dimensional steady inviscid Adiabatic flow :

The relation between enthalpy and velocity is continuous.

0ududh

udh

du 1

Summary

2

2

1

1

MA

u

dA

dp

1

12

MA

u

dA

du

udp

du

1

udh

du 1

p

dp

u

duM

12

p

dp

A

dA

M

M

1

12

2

d

p

dp

d

A

dA

M

M

12

2

Subsonic Nozzle Subsonic Diffuser

dA < 0 & M <1

So, du > 0 & dp <0

dA > 0 & M <1

So, du < 0 & dp>0

A

dA

Mu

du

1

12

T

dT

A

dA

M

M

1

12

2

Supersonic Diffuser Supersonic Nozzle

dA < 0 & M >1

So, du < 0 & dp >0

dA > 0 & M >1

So, du >0 & dp<0

p

dp

u

duM

12

p

dp

A

dA

M

M

1

12

2

A

dA

Mu

du

1

12

d

p

dp

d

A

dA

M

M

12

2 T

dT

A

dA

M

M

1

12

2

Generation of High Pressure from Supersonic velocity

u

du

Mp

dp2

p

dp

M

M

A

dA

11

2

2

p

dpd

1

p

dp

T

dT

1

An Ideal Diffuser at Design Conditions

p1p2d

pthroat

p

p1

p*

p2d

Generation of Supersonic Velocity from Rest

p

dpM

u

du

2

p

dp

M

M

A

dA

11

2

2

p

dpd

1

p

dp

T

dT

1

RAMJET Engine

Capacity of A Cross Section : An implicit Model

Mass flow rate through any cross section of area A

)()()( xuxAxm

With a condition that sonic velocity occurs at throat !

thoratthroatthroat CAm

throatthroat

throatthorat RT

pC

Stagnation Temperature for theAdiabatic Flow of a Calorically Perfect Gas

• Consider an adiabatic flow field with a local gas Temperature T(x), pressure p(x), and a velocity V(x)

x

T(x)

V(x)

p(x)

• Since the Flow is adiabatic

constant2

)()(

2

)()(

22

xu

xTCxu

xh p

y

To(y)

V(y)=0

po(y)

Introduce an obstruction in the inviscid flow field :

• This obstruction generates a location y, within this flow field where the gas velocity is reduced to zero.

• Since the Flow is adiabatic

ppp C

xuT(x)TTC

xuxTC

2

)(

2

)()(

2

00

2

x

T(x)

V(x)

p(x)

12

)()(

2

0

R

xuxTT

20

2

0 )(2

1)(

12

)()( xu

RxTT

R

xuxTT

22

0 )(2

11)(

)(

)(

2

11)( xMxT

xRT

xuxTT

Holds anywhere within an adiabatic Flow field

In general for an adiabatic Flow Field the Stagnation Temperature is defined by the relationship

T0

T1

1 2

M 2

Stagnation Temperature is constant throughout an adiabatic flow field.

• T0 is also sometimes referred to at Total Temperature

• T is sometimes referred to as Static Temperature

• Stagnation temperature is a measure of the Kinetic Energy of the flow Field.• Largely responsible for the high Level of heating that occurs on high speed aircraft or reentering space Vehicles …

T0

T1

1 2

M 2

p0

p

T0

T

1

1 1

2M 2

1

“stagnation” (total) pressure : Constant throughout Isentropic flow field.

• Similarly Stagnation density for isentropic flow field is

0

T0

T

1

1 1

2M 2

1

11

Stagnation Properties of Isentropic Flow

T0

T1

1 2

M 2

p0

p

T0

T

1

1 1

2M 2

1

0

T0

T

1

1 1

2M 2

1

11

What was Stagnation Temperature At Columbia Breakup

Loss Of Signal at:61.2 km altitude~18.0 Mach Number

T∞ ~ 243 K

T0

T1

1 2

M 2

Ideal gas Variable Properties Real gas

Capacity of A Cross Section

Mass flow rate through any cross section of area A

)()()( xuxAxm

With a condition that sonic velocity occurs at throat !

thoratthroatthoat CAm

thoatthorat

thoatthorat RT

pC

Calorically perfect gas:

1

1

20

12

1

1

00

)(2

11

1)()(

2

11

)()(

xMxxM

xT

T

x

)()()(

21

1

11

1

20 xuxA

xMm

T0

T1

1 2

M 2

2

0

)(2

11

)(xM

TxT

)()()()(

21

1

11

1

20 xcxMxA

xMm

RTxMxAxM

m

)()()(

21

1

11

1

20

2

0

1

1

20

)(2

11

)()()(

21

1

1

xM

TRxMxA

xMm

)()()(

21

1

12

1

1

1

200 xMxA

xMRTm

)()(

)(2

11

1

12

1

2

00 xMxA

xM

RTm

12

1

2

00

)(2

11

)()(

xM

xMxARTm

12

1

20

00

)(2

11

)()(

xM

xMxA

RT

pRTm

12

1

20

0

)(2

11

)()(

xM

xMxA

T

p

Rm

Specific Mass flow Rate

Mass flow rate per unit area of cross section:

12

1

20

0

)(2

11

)(

)(

xM

xM

T

p

RxA

m

Design of Converging Diverging Nozzles

P M V SubbaraoAssociate Professor

Mechanical Engineering DepartmentI I T Delhi

From the Beginning to the Peak or Vice Versa….

Quasi-One-Dimensional Flow

Distinction Between True 1-D Flow and Quasi 1-D Flow

• In “true” 1-D flow Cross sectional area is strictly constant• In quasi-1-D flow, cross section varies as a Function of the longitudinal coordinate, x• Flow Properties are assumedconstant across any cross-section• Analytical simplification very useful for evaluating Flow properties in Nozzles, tubes, ducts, and diffusersWhere the cross sectional area is large when compared to length

Specific Mass flow Rate

Mass flow rate per unit area of cross section:

12

1

20

0

)(2

11

)(

)(

xM

xM

T

p

RxA

m

Maximum Capacity of A Nozzle

12

1

20

0

)(2

11

)(

)(

xM

xM

T

p

RxA

m

• Consider a discontinuity at throat “choked-flow” Nozzle … (I.e. M=1 at Throat)

• Then comparing the massflow /unit area at throat to some other station.

12

1

20

0

)(2

11

)(

)(

xM

xM

T

p

RxA

m

12

10

0

21

1

1

T

p

RA

m

throat

Take the ratio of the above:

12

10

0

12

1

20

0

21

1

1

)(2

11

)(

)(

T

p

R

xM

xM

T

p

R

xA

Athroat

12

1

12

1

2

21

1

1

)(2

11

)(

)(

xM

xM

xA

Athroat

12

1

2

12

1

)(2

11

21

)()(

xM

xMxA

Athroat

12

1

2)(2

11

1

2

)(

1)(

xMxMA

xA

throat

12

1

2*

)(2

11

1

2

)(

1)(

xMxMA

xA

Design Analysis

12

1

2*

)(2

11

1

2

)(

1)(

xMxMA

xA

For a known value of Mach number, it is easy to calculate area ratio. Throat area sizing is the first step in the design.

If one needs to know the Mach number distribution for a given geometric design!

Find the roots of the non-linear equation.

Typical Design Procedure

• The Space Shuttle Main Engines burn LOX/LH2 for propellants with A ratio of LOX:LH2 =6:1

• The Combustor Pressure, p0 is 20.4 Mpa, combustor temperature, T0 is 3300K.

• Decide throat diameter based on the requirement of thrust.

• What propellant mass flow rate is required for choked flow in the Nozzle?

• Assume no heat transfer through Nozzle no frictional losses.

• Combustion product is water vapor.

Space Shuttle Main Engines

12

10

0

21

1

1

T

p

RA

m

throat

Specifications of SSME

• Specific Impulse is a commonly used measure of performanceFor Rocket Engines,and for steady state-engine operation is definedAs:

I sp 1

g0

Fthrust•

m propellant

g0 9.806m

sec2(mks)

• At 100% Throttle a SSSME has the Following performance characteristics

Fvac = 2298 kNt

Fsl = 1600 kNt

Ispvac = 450 sec.

SEA Level Performance

One needs to know the Mach number distribution for a given geometric design!

Find the roots of the non-linear equation.

Numerical Solution for Mach Number Caluculation

• Use “Newton’s Method” to extract numerical solution

• At correct Mach number (for given A/A*) …

F(M ) 0

F(M ) 1

M

2

1

1 1

2M 2

1

2 1

A

A*

• Define:

• Expand F(M) is Taylor’s series about some arbitrary Mach number M(j)

F(M ) F(M ( j ) ) F

M

( j )

M M ( j ) 2F

M 2

( j )

M M ( j ) 2

2 ...O M M ( j ) 3

• Solve for M

M M ( j )

F(M ) F(M ( j ) )

2F

M 2

( j )

M M ( j ) 2

2 ...O M M ( j ) 3

F

M

( j )

• From Earlier Definition , thusF(M ) 0

M M ( j )

F(M ( j ) )

2F

M 2

( j )

M M ( j ) 2

2 ...O M M ( j ) 3

F

M

( j )

• if M(j) is chosen to be “close” to M M M ( j ) 2 M M ( j )

And we can truncate after the first order terms with “little”Loss of accuracy

Still exact expression

• First Order approximation of solution for M

• However; one would anticipate that

“Hat” indicates that solution is no longer exact

M^

M ( j ) F(M ( j ) )

F

M

( j )

M M^

M M ( j )

“estimate is closer than original guess”

• And we would anticipate that

“refined estimate” …. Iteration 1

M^^

M^

F(M

^

)F

M

|M^

M M^^

M M^

• If we substitute back into the approximate expressionM^

• Abstracting to a “jth” iteration

Iterate until convergencej={0,1,….}

M^

( j1) M^

( j ) F(M

^

( j ) )F

M

|( j )

1

M^

( j1)

2

1

1 1

2M

^

( j1)

2

1

2 1

A

A*

A

A*

• Drop from loop when

Plot Flow Properties Along Nozzle Length

• A/A*

• Mach NumberM

^

( j1) M^

( j ) F(M

^

( j ) )F

M

|( j )

• Temperature T (x) T0

1 1

2M (x)2

T0 = 3300KTthroat = 2933.3 K

• Pressure

P0 = 20.4Mpa Pthroat = 11.32 MPa

P(x) P0

1 1

2M (x)2

1

daa AuAu 222111

dda AuApAuApa 2

22221

2111 21

Nozzle at Off Design Exit Pressure

apa TTc

uu,21

222

221

p1p2d

p2a > p2d

pthroat

Nozzle at Off Design Exit Pressure

p1

p1 > p2a > pthroat

pthroat

p

p1 p2a

p*

P*2a

p2d