1. Turbomachinery Lecture 3 Compressibility Isentropic
Area, Mass Flow Functions
C-D Nozzle

2. Turbomachinery Definition:
A turbomachine transfers energy to or from a fluid moving continuously through a casing by dynamic action of a rotor and by flow conditioning of a stator.
Works on a fluid to produce power or flow (and pressure rise)
Adds energy to fluid Pump or Compressor
Fan: pressure rise up to lbf/in2
Blower: pressure between lbf/in2
Compressor: pressure rise > 40 lbf/in2
Extracts energy from fluid Turbine
Pressure changes due to motion of parts or displacement of boundaries

3. Compressible Flow
Density varies making continuity & momentum more difficult to solve.
because  varies with velocity.
Also, can't integrate Bernoulli directly
Compressible flow problems can be solved iteratively using continuity, state et. al. α C

4. Compressible Flow Example: m = 50 lb/sec A = 200 sq.in.
P0 = 14.7 psia  = 30
T0 = 519 R
Guess C = ft/sec

5. Compressible Flow
Pressure

6. Compressible Flow
Density can now be found from state:

7. Compressible Flow
Mass Flow:
note:
19%>

8. Compressible Flow Mach Number Functions:
Easily calculated & clarify physics
Mach number & acoustic speed are critical concepts!

9. Compressible Flow

10. Compressible Flow
Using isentropic relation between pressure & temperature derivatives:
Use adiabatic state law

11. Compressible Flow
Using equation of state, acoustic speed in an ideal gas is [from kinetic theory]:
By definition Mach Number is:

12. Compressible Flow
Static & Total properties as functions of Mach number:

13. Compressible Flow – Critical Velocity
What does subscript * mean? It means value of variable when M=1 [sonic]
Vcr is only function of gas [] and stagnation props.

14. Compressible Flow
The relation between static & stagnation properties is isentropic. Then:

15. Compressible Flow
The relation between compressible and Bernoulli [B-p.55]

16. Compressible Flow Relationships
Mass Flow parameter [=0]

17. Compressible Flow Relationships
Area-Mach number differential relation
Area-Mach number integral relation
More on next chart

18. Compressible Flow Relationships
What does subscript * mean?
For all flow variables it means value of variable when M=1 [sonic]
For area A* this is reference area for choking flow [M=1]
Note this area is a minimum or throat
More on next chart

19. Compressible Flow Relationships
Compressible
Aero Calculator
calcbody2
Flow textbooks

20. Compressible Flow Relations

21. Of interest
here
Of interest
here

23. Over-expanded

26. Compressible Flow Examples

27. Compressible Flow Examples

28. Compressible Flow
Mass Flow Parameters:

29. Compressible Flow Relationships
Mass flow parameters
Note:
FPo, FPs are similar,
but different f[M] powers

30. Compressible Flow Relationships
Mass flow parameters
How to get more mass flow, i.e. greater thrust, more power?

31. Compressible Flow Relationships
Mass flow parameters in English units
m in lbm/sec
p0A in lbf [spatial dimensions cancel]
T0 in degs. Rankine

32. Compressible Flow Relationships
Mass flow parameters in SI units
m in kg/sec
p0A in Newtons [spatial dimensions cancel]
T0 in degs. Kelvin

33. Compressible Flow Examples

34. Example

35. Static Pressure Mass Flow Parameter
Defining: FP = Flow parameter=f(M)
For Air
Can be inverted

36. Total Pressure Mass Flow Parameter
Introducing P0:
No explicit solution for M
FPs is single valued, FPo is not
FPo max = for =1.4
FPo max always at M=1

37. Calculate FPo From Previous Example: m = 50 lb/sec A = 200 sq.in.
P0 = 14.7 psia  = 30
T0 = 519 R
Rearrange FPo

38. Mass Flow Parameters
Be careful: FPs single valued, FPo double values

39. Total Pressure Mass Flow Parameter
Consider FPt:
For fixed , a fixed value of
produces the same Mach number - regardless of the level of pressure, temperature or molecular weight (R).

40. Total Pressure Mass Flow Parameter
Defines common flow parameters.
Valid for flow with one gas.
Corrected flow.

41. Other Parameters [Covered in Lecture 4]
Ideal gas equation for Mach number leads to speed parameters, also for a single gas.
Speed parameter
Corrected speed

42. Significance of Flow & Speed Parameters
A device operating at:
same speed parameter and flow parameter has
same Mach numbers, velocity diagrams, flow angles etc,
regardless of level of physical speed, pressure & temperature.

43. Flow and Speed Parameters
Conditions: same gas, high Reynolds number, same clearances, and same 
Speed and Flow parameters are used for turbine maps

44. Corrected Flow & Speed Parameters
Corrected Flow and Corrected speed used for compressor maps

45. Flow Parameter Again, Consider FP0:
Unlike P, T & R;  cannot be "corrected".
Changing , changes relation between FP0 and Mach number!

46. Flow Parameter
Message: More complex gasses choke at a lower Mach number