Design of a Multi-Stage Compressor Motivation: Market research has shown the need for a low-cost turbojet with a take-off thrust of 12,000N. Preliminary studies will show that a single-spool all-axial flow machine is OK, using a low pressure ratio and modest turbine inlet temperatures to keep cost down. Problem: Design a suitable compressor operating at sea-level static conditions with compressor pressure ratio = 4.15 air mass flow = 20 kg/s turbine inlet temperature = 1100K Assume: Pamb = 1.01 bar, Tamb = 288 K Utip = 350 m/s Inlet rhub / rtip = 0.5 Compressor has no inlet guide vanes Mean radius is constant Polytropic efficiency = 0.90 Constant axial velocity design No swirl at exit of compressor

Steps in the Meanline Design Process 1) Choice of rotational speed and annulus dimensions 2) Determine number of stages, using assumed efficiency 3) Calculate air angles for each stage at the mean radius - meanline analysis 4) Determine variation of the air angles from root to tip - radial equilibrium 5) Investigate compressibility effects 6) Select compressor blading, using experimentally obtained cascade data or CFD 7) Check on efficiency previously assumed 8) Estimate off-design performance

Compressor Meanline Design Process Steps Choose Cx1 and rH/rT to satisfy m and keep Mtip low and define rT Select N from rT and UT Compute T0 across compressor and all exit flow conditions [keep rm same through engine] Estimate T0 for first stage from inlet condtions [Euler and de Haller] Select number of stages  T0comp / T0stage …..

Step 1- Choice of Rotational Speed & Annulus Dimensions Construct table of inlet / exit properties and parametric study of c1x vs. tip Mach number [next chart] Chose c1x from spread sheet to avoid high tip Mach numbers and stresses Calculate 1 from inlet static pressure and temperature With mass flow = 20 kg/s and rhub/rtip = 0.5 and compute rotational speed and tip Mach number Choose N = 250 rev/sec or 15,000 RPM and rhub/rtip = 0.5, calculated rhub, rtip, rmean

Calculate Tip Radius and Rotational Speed Drive choice by compressor inlet conditions

Compressor Meanline Design c1x chosen to avoid high tip Mach numbers and stresses Calculate compressor inlet details (1) Given: m, Utip, p01, T01, Pr, poly calculate compressor exit details including Aexit, h exit (2) Calculate 1 from U, Cx Calculate W2=0.72W1, 2 from Cx, W2 From Euler, 1 and 2, calculate T0-12

Step 3-Choose Number of Stages Given poly and T0out/T0in  T0 = T0out -T0in, so the number of stages is DT0 compressor / DT0stage = 164.5/28 = 5.9 To be conservative (account for losses, i.e. ha<1), Choose 7 stages Recalculate the DT0stage = 164.5/7 = 23.5 Calculate 1st stage temperature ratio is T0 ratio and pressure ratio

Step 3 - Calculate Velocity Triangles of 1st Stage at Mean Radius [Rm] Use Euler Turbine Equation to recalculate 2 and check de Haller Calculate 1st stage rotor exit conditions [2] Calculate Reaction [R1=0.84, high] Consider the stator of the 1st stage, assume “repeating stage”, then inlet angle to stator [α1=0] is absolute air angle coming out of rotor [α3] and, exit absolute angle of stator is inlet absolute angle of rotor Constant Cx U C1 W2 W1 C2

Velocity Triangles of 1st Stage Using “Repeating Stage” Assumption Notice that the velocity triangles are not “symmetric” between the rotor and stator due to the high reaction design of the rotor. The rotor is doing most of the static pressure (temperature) rise. Cx1=150 a1=0 W = C - U b1=60.64 U=- WU1 =266.6 W1=305.9 STATOR Cx3=150 a3=0 b3=60.64 C2=174.21 a2=30.57 U=266.6 CU2=88.6 W3=305.9 Cx2=150 ROTOR U=266.6 b2=49.89 WU2=178.0 W2=232.77

Stage Design Repeats for Stages 2-7 Then the mean radius velocity triangles “essentially” stay the same for stages 2-7, provided: mean radius stays constant hub/tip radius ratio and annulus area at the exit of each stage varies to account for compressibility (density variation) stage temperature rise stays constant reaction stays constant If, however, we deviate from the “repeating stage” assumption, we have more flexibility in controlling each stage reaction and temperature rise.

Non- “Repeating Stage” Design Strategy Instead of taking a constant temperature rise across each stage, we could reduce the stage temperature rise for the first and last stages of the compressor and increase it for the middle stages. This strategy is typically used to: reduce the loading of the first stage to allow for a wide variation in angle of attack due to various aircraft flight conditions reduce the turning required in the last stage to provide for zero swirl flow going into the combustor With this in mind, lets change the work distribution in the compressor to:

Velocity Triangles of 1st Rotor Using “Non-Repeating Stage” Assumption Cx1=150 Notice that the velocity triangles are not “symmetric” due to the high reaction design of the rotor. Also, there is swirl now leaving the stator. a1=0 W = C - U b1=60.64 U=- WU1 =266.6 C3=153.56 W1=305.9 STATOR a3=12.36 Cx3=150 CU3=32.87 b3=57.31 C2=167.87 U=266.6 a2=26.68 CU2=75.38 W3=277.73 Cx2=150 ROTOR U=266.6 b2=51.89 WU3=233.77 WU2=191.22 W2=242.03

Velocity Triangles of 1st Stage Using “Repeating Stage” Assumption Notice that the velocity triangles are not “symmetric” between the rotor and stator due to the high reaction design of the rotor. The rotor is doing most of the static pressure (temperature) rise. Cx1=150 a1=0 W = C - U b1=60.64 U=- WU1 =266.6 W1=305.9 STATOR Cx3=150 a3=0 b3=60.64 C2=174.21 a2=30.57 U=266.6 CU2=88.6 W3=305.9 Cx2=150 ROTOR U=266.6 b2=49.89 WU2=178.0 W2=232.77 13 13

Design of 2nd Stage Stator & 3rd Stage Rotor Design of the 2nd stage stator and 3rd stage rotor can be done in the same manner as the 1st stage stator and 2nd stage rotor. A choice of 50% reaction and a temperature rise of 25 degrees for the 3rd stage will lead to increased work by the stage but a more evenly balanced rotor/stator design. The velocity triangle of the stator will be a mirror of the rotor. This stage design will then be repeated for stages 4 - 6.

Design of 2nd Stage Stator The pressure ratio for the 2nd stage design with a temperature change, DT0 = 25 is: So P03= P02= 1.248 (1.279) = 1.596 bar and T03= 308+25=333 0K Now we must choose a value of a3 leaving the 2nd stage stator that provides for the desired Reaction and Work in the 3rd stage using a similar technique as previously used.

Design of 2nd Stage Stator & 3rd Stage We can change a3 so that there is swirl going into the third stage and thereby reduce the reaction of our second stage design. If we design the third stage to have a reaction of 0.5, then from the equation for reaction: And if we design the third stage to a temperature rise of 25 0, the Euler’s equation: Which can be solved simultaneously for b1and b2

Velocity Triangles of 2nd Stage W = C - U Cx3=150 CU3=32.87 b3=57.31 C3=172.99 U=266.6 a3=29.88 Cx3=150 CU3=86.18 W3=277.73 STATOR b3=50.26 U=266.6 WU3=233.77 C2=196.59 W3=234.63 a2=40.27 CU2=127.07 Cx2=150 ROTOR WU3=180.42 U=266.6 b2=42.92 WU2=139.53 W2=204.86 Notice that the velocity triangles are not “symmetric” for the second stage due to 70%reaction design but will be for 3rd stage (50% reaction).

Design of Stages 4-6 The velocity triangles of stages 4 through 6 will essentially be repeats of stage 3 since all have a 50% reaction and a temperature rise of 25 degrees. Stagnation and static pressure as well as stagnation and static temperature of these stages will increase as you go back through the machine. As a result, density will also change and will have to be compensated for by changing the spanwise radius difference (area) between the hub and tip (i.e. hub/tip radius ratio)

Velocity Triangles of Stages 3 - 6 W = C - U Cx3=150 CU3=86.18 C3=172.99 b3=50.26 a3=29.88 Cx3=150 CU3=86.18 U=266.6 STATOR b3=50.26 W3=234.63 U=266.6 W3=234.63 WU3=180.42 C2=234.63 CU2=180.42 WU3=180.42 a2=50.26 Cx2=150 ROTOR U=266.6 b2=29.88 WU2=86.18 W2=172.99 Notice that the velocity triangles are “symmetric” due to the 50%reaction design.

Stage 7 Design For stage 7, we have P01= 3.65 and T01 = 433. For this our 7-stage compressor design we have P0 exit = 4.15 * 1.01 = 4.19 bar T0 exit = 288.0 (4.15)0.3175 = 452.5 0K If we assume a Reaction = 0.5 for the 7th stage: or:

Stage 7 Design And from: or from symmetry of the velocity triangles for 50% reaction: Note that the absolute angles going into stage 7 have changed from those computed for stages 3 - 6 and that the exit absolute air angle leaving the compressor is 32.770. This means that a combustor pre-diffuser is required to take all of the swirl out of the flow prior to entering the combustor.

Summary of Compressor Design

Conditions in Compressor

Hub & Tip Radii Distribution - Flow Path Area