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Stagnation Properties . P M V Subbarao Professor Mechanical Engineering Department I I T Delhi. Capacity of A Resource…. 1. Stagnation Properties of Isentropic Flow. What was Stagnation Temperature At Columbia Breakup. Loss Of Signal at: 61.2 km altitude ~18.0 Mach Number.

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stagnation properties

Stagnation Properties

P M V Subbarao

Professor

Mechanical Engineering Department

I I T Delhi

Capacity of A Resource…..

what was stagnation temperature at columbia breakup
What was Stagnation Temperature At Columbia Breakup

Loss Of Signal at:

61.2 km altitude

~18.0 Mach Number

T∞ ~ 243 K

slide4

Ideal & Calorically perfect Gas

Ideal Gas with Variable Properties

Real Gas with Variable Properties

capacity of a cross section
Capacity of A Cross Section

Mass flow rate through any cross section of area A

Maximum Capacity is obtained when sonic velocity occurs at throat !

specific mass flow rate
Specific Mass flow Rate

Mass flow rate per unit area of cross section:

design of supersonic intake nozzle

Design of Supersonic Intake / Nozzle

P M V Subbarao

Associate Professor

Mechanical Engineering Department

I I T Delhi

From the Beginning to the Peak or Vice Versa….

slide13

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 assumed

constant across any cross-section

• Analytical simplification very useful for evaluating Flow properties in Nozzles, tubes, ducts, and diffusers.

Where the cross sectional area is large when compared to length

specific mass flow rate14
Specific Mass flow Rate

Mass flow rate per unit area of cross section:

maximum capacity of an intake nozzle
Maximum Capacity of An Intake/Nozzle

• 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.

design analysis
Design Analysis

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 we know the details of the resource/requirements, we can calculate the size of throat.

cryogenic rocket engines
Cryogenic Rocket Engines

A ratio of LO2:LH2 =6:1

T0 = 3300K.

P0 = 20.4 Mpa

specifications of a rocket engine
Specifications of A Rocket Engine

• Specific Impulse is a commonly used measure of performance

For Rocket Engines,and for steady state-engine operation is defined

As:

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

Fvacuum = 2298 kNt

Ispvacuum = 450 sec.

Fsea level = 1600 kNt

design procedure
Design Procedure

Select a technology : Isp & Fthrust

sea level performance

SEA Level Performance

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

Find the roots of the non-linear equation.

slide25

Numerical Solution for Mach Number Caluculation

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

• Define:

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

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

slide27

• From Earlier Definition , thus

Still exact expression

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

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

Loss of accuracy

slide28

• First Order approximation of solution for M

“Hat” indicates that solution is no longer exact

• However; one would anticipate that

“estimate is closer than original guess”

slide29

• If we substitute back into the approximate expression

• And we would anticipate that

“refined estimate” …. Iteration 1

slide30

• Abstracting to a “jth” iteration

Iterate until convergence

j={0,1,….}

• Drop from loop when

slide33

• Temperature

T0 = 3300K

Tthroat = 2933.3 K

slide34

• Pressure

P0 = 20.4Mpa

Pthroat = 11.32 MPa

operating characteristics of nozzles

Operating Characteristics of Nozzles

P M V Subbarao

Professor

Mechanical Engineering Department

I I T Delhi

Realizing New Events of Physics…….

converging nozzle
Converging Nozzle

pb = Back Pressure

Design Variables:

Outlet Condition:

p0

pb

designed exit conditions
Designed Exit Conditions

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

profile of the nozzle
Profile of the Nozzle

At design Conditions:

remarks on isentropic nozzle design
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 nozzle41
Converging Nozzle

p0

Pb,critical

operational characteristics of nozzles
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

pb,critical<pb2<p0

pb,critical<pb3<p0

Pressure Distribution in Under Expanded Nozzle

pb=p0

p0

pb,critical<pb1<p0

Pb,critical

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

convergent divergent nozzle with high back pressure
Convergent-Divergent Nozzle with High Back Pressure
  • When pbis very nearly the same as p0the 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 pressure52
Convergent-Divergent Nozzle with High Back Pressure

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

pthroat> p*

slide56

At exit with high back pressure pb

At throat with high back pressure pb

slide57

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