Operating Characteristics of Nozzles

P M V SubbaraoProfessor

Mechanical Engineering DepartmentI I T Delhi

From Takeoff to cruising ……Realizing New Events of Physics…….

Converging Nozzle

p0

pb

pb = Back Pressure

Design Variables: 00 ,, Tpm

Outlet Condition:

exitexitb MorAorp

Designed Exit Conditions

121

00

2

11

exit

exitexit

MT

T

p

p

12

1

20

0

21

1

exit

exitexit

M

MA

T

p

Rm

Under design conditions the pressure at the exit plane of the nozzle is applied back pressure.

121

00

2

11

exit

exitb

MT

T

p

p

Profile of the Nozzle 12

1

2

2

21

1

)(2

11

)(

)(

exit

exit

exit M

xM

xM

M

A

xA

1

2

2

)(12

12)(

xM

M

p

xp exit

b

1

2

2

)(12

12)(

xM

M

p

xp exit

exit

At design Conditions:

Full Capacity Convergent Nozzle

1

2, )(12

1)(

xMp

xp

criticalb

12

1

21

)(2

11

)(

1)(2

,

xM

xMA

xA

criticalexit

Remarks on Isentropic Nozzle Design

• Length of the nozzle is immaterial for an isentropic nozzle.

• Strength requirements of nozzle material may decide the nozzle length.

• Either Mach number variation or Area variation or Pressure variation is specified as a function or arbitrary length unit.

• Nozzle design attains maximum capacity when the exit Mach number is unity.

Converging Nozzle

p0

Pb,critical

1

,

0

2

1

criticalbp

p

1

0, 1

2

pp criticalb

Operational Characteristics of Nozzles

• A variable area passage designed to accelerate the a gas flow is considered for study.

• The concern here is with the effect of changes in the upstream and downstream pressures

• on the nature of the inside flow and • on the mass flow rate through a nozzle.• Four different cases considered for analysis are:• Converging nozzle with constant upstream conditions.• Converging-diverging nozzle with constant upstream conditions.• Converging nozzle with constant downstream conditions.• Converging-diverging nozzle with constant downstream

conditions.

Pressure Distribution in Under Expanded Nozzle

p0

Pb,critical

pb=p0

pb,critical<pb1<p0

pb,critical<pb2<p0

pb,critical<pb3<p0

At all the above conditions, the pressure at the exit plane of nozzle, pexit = pb.

Variation of Mass Flow Rate in Exit Pressure

0p

pb

0p

pe

1

1

0

,

p

p criticalb

0

,

p

p criticale

Variation of in Exit Pressure

0p

pb

0p

pe

1

1

0

,

p

p criticalb

0

,

p

p criticale

Variation of in Mass Flow Rate

0p

pb

m

10

,

p

p criticalb

chokedm

Low Back Pressure Operation

0

)(

p

xp

0

*

p

p

00 p

p

p

p bexit

Convergent-Divergent Nozzle Under Design Conditions

Convergent-Divergent Nozzle with High Back Pressure

p*< pb1<p0

pthroat> p*

Convergent-Divergent Nozzle with High Back Pressure

• When pb is very nearly the same as p0 the flow remains subsonic throughout.

• The flow in the nozzle is then similar to that in a venturi.

• The local pressure drops from p0 to a minimum value at the throat, pthroat , which is greater than p*.

• The local pressure increases from throat to exit plane of the nozzle.

• The pressure at the exit plate of the nozzle is equal to the back pressure.

• This trend will continue for a particular value of back pressure.

Convergent-Divergent Nozzle with High Back Pressure

At all these back pressures the exit plane pressure is equal to the back pressure.

pthroat> p*

0

2

2

)()( TC

xuxTC pp

12

)(

1

)( 20

22

cxuxc

12

1

0

12

0

2

1

00

)()()()(

x

p

xp

T

xT

c

xc

)(1

2

1

)(

12)( 22

0

2202 xcc

xccxu

20

2202 )(

11

2)(

c

xccxu

1

0

202 )(

11

2)(

p

xpcxu

1

0

02 )(1

1

2)(

p

xpRTxu

1

00

02 )(1

1

2)(

p

xppxu

2/1

1

00

0 )(1

1

2)(

p

xppxu

2/1

1

00

0

00

)(1

1

2)(

)(

p

xppxA

xm

)()()( xuxAxm

)()()(

00 xuxA

xm

2/1

1

00

0

1

00

)(1

1

2)(

)(

p

xppxA

p

xpm

At exit with high back pressure pb

2/1

1

00

0

1

00 1

1

2

p

ppA

p

pm exit

exitb

At throat with high back pressure pb

2/1

1

00

0*

1

00 1

1

2

p

ppA

p

pm tt

2/1

1

0

*

1

0

2/1

1

0

1

0

11

p

pA

p

p

p

pA

p

p ttbexit

b

•For a given value of high back pressure corresponding throat pressure can be calculated. •As exit area is higher than throat area throat pressure is always less than exit plane pressure.•An decreasing exit pressure produces lowering throat pressure

Variation of Mass Flow Rate in Exit Pressure

0p

pb

0p

pe

1

1

0

,

p

p criticalb

0

,

p

p criticale

Variation of in Mass Flow Rate

0p

pb

m

10

,

p

p criticalb

chokedm

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