1. One Dimensional Flow of Blissful Fluid -III
B.GOVINDHARAJAN
J.PARTHIBAN
Always Start with simplest Inventions……..

2. Differential Form of Momentum Equation
One dimensional steady inviscid flow :
The relation between pressure and velocity is continuous.

3. Differential Form of Energy Equation
One dimensional steady inviscid Adiabatic flow :
The relation between enthalpy and velocity is continuous.

4. Summary

5. Subsonic Nozzle
Subsonic Diffuser
dA < 0 & M <1
So, du > 0 & dp <0
dA > 0 & M <1
So, du < 0 & dp>0

6. Supersonic Diffuser
Supersonic Nozzle
dA < 0 & M >1
So, du < 0 & dp >0
dA > 0 & M >1
So, du >0 & dp<0

7. Generation of High Pressure from Supersonic velocity

8. An Ideal Diffuser at Design Conditions
pthroat
p2d p1 p1
p2d p p*

9. Generation of Supersonic Velocity from Rest

10. RAMJET Engine

11. Capacity of A Cross Section : An implicit Model
Mass flow rate through any cross section of area A
With a condition that sonic velocity occurs at throat !

12. Stagnation Temperature for the Adiabatic 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)
• Since the Flow is adiabatic

13. 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

14. Holds anywhere within an adiabatic Flow field

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

16. • Similarly Stagnation density for isentropic flow field is
• 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 …
“stagnation” (total) pressure : Constant throughout Isentropic flow field.
• Similarly Stagnation density for isentropic flow field is 1

17. Stagnation Properties of Isentropic Flow1

18. What was Stagnation Temperature At Columbia Breakup
Loss Of Signal at:
61.2 km altitude
~18.0 Mach Number
T∞ ~ 243 K

19. Ideal gas Variable Properties Real gas

20. Capacity of A Cross Section
Mass flow rate through any cross section of area A
With a condition that sonic velocity occurs at throat !

21. Calorically perfect gas:

25. Specific Mass flow Rate
Mass flow rate per unit area of cross section:

26. Design of Converging Diverging Nozzles
B.GOVINDHARAJAN
J.PARTHIBAN
From the Beginning to the Peak or Vice Versa….

27. Quasi-One-Dimensional Flow

28. 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

29. Specific Mass flow Rate
Mass flow rate per unit area of cross section:

30. Maximum Capacity of A 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.

31. Take the ratio of the above:

34. For a known value of Mach number, it is easy to calculate area ratio.
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 one needs to know the Mach number distribution for a given geometric design!
Find the roots of the non-linear equation.

35. 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 3300 K.
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.

36. Space Shuttle Main Engines

37. Specifications of SSME
• 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 SSSME has the Following performance characteristics
Fvac = kNt
Fsl = 1600 kNt
Ispvac = 450 sec.

38. SEA Level Performance
One needs to know the Mach number distribution for a given geometric design!
Find the roots of the non-linear equation.

39. 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)

40. • Solve for M

41. • 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

42. • 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”

43. • If we substitute back into the approximate expression
• And we would anticipate that
“refined estimate” …. Iteration 1

44. • Abstracting to a “jth” iteration
Iterate until convergence
j={0,1,….}
• Drop from loop when

45. Plot Flow Properties Along Nozzle Length
• A/A*

46. • Mach Number

47. • Temperature
T0 = 3300K
Tthroat = K

48. • Pressure
P0 = 20.4Mpa
Pthroat = MPa

49. Nozzle at Off Design Exit Pressure
pthroat
p2d p1
p2a > p2d

50. Nozzle at Off Design Exit Pressure
pthroat p1
p1 > p2a > pthroat p1
p2a p
P*2a p*
p2d