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 flowing continuously through a casing by the dynamic action of a rotor and by the 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 1 lbf/in 2 –Blower: pressure between lbf/in 2 –Compressor: pressure rise > 40 lbf/in 2 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.

4 Example : m = 50 lb/secA = 200 sq.in. P 0 = 14.7 psia  = 30  T 0 = 519  R GuessC = ft/sec Compressible Flow

5 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 Flow textbooks - - Compressible - Aero Calculator - calcbody2

20 Compressible Flow Relations

21 Of interest here Of interest here

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Over-expanded 23

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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 units –m in lbm/sec –p 0 A in lbf [spatial dimensions cancel] –T 0 in degs. Rankine –A is sometimes frontal area Acos 

32 Compressible Flow Examples

33 Example

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

35 Total Pressure Mass Flow Parameter Introducing P 0 : No explicit solution for M FP s is single valued, FPo is not FPo max = for  =1.4 FPo max always at M=1

36 Calculate FPo From Previous Example: m = 50 lb/secA = 200 sq.in. P 0 = 14.7 psia  = 30  T 0 = 519  R Rearrange FPo

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

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

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

40 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

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

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

43 Corrected Flow & Speed Parameters Corrected Flow and Corrected speed used for compressor maps

44 Flow Parameter Again, Consider FP 0 : Unlike P, T & R;  cannot be "corrected". Changing , changes relation between FP 0 and Mach number!

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

46 Example Solution to Mass Flow Parameter

47 Area Ratio A/A* is flow area / flow area at M = 1.0