Turbomachinery Lecture 2b Efficiency Definitions

Gas-Turbine Brayton Cycle Most Gas Turbines Use Ideal Brayton Cycle as Basis for Design Isentropic compression (2 to 3) Constant pressure heat addition (3 to 4) Isentropic expansion (4 to 9) Constant pressure heat rejection (9 to 2)

Gas-Turbine Brayton Cycle 3 2 4 9 Compressor Combustor Turbine Ambient For constant (or nearly constant) velocity Neglect fuel mass flow effect

Gas-Turbine Brayton Cycle 3 2 4 9 Compressor Combustor Turbine Ambient Gas-Turbine Brayton Cycle

Compressor Adiabatic (Isentropic) Efficiency Poout h02 Usually in terms of stagnation properties, but in centrifugal and industrial machines  inlet use Po, exit Ps h02i V2/2 If inlet/outlet K.E. small, Poin h01 s

Compressor Efficiency Compressor Efficiency is a Function of Compressor Pressure Ratio Pressure Ratio of Each Stage Isentropic Efficiency of Each Stage Consider Special Case Where Each Stage Pressure Ratio and Each Stage Efficiency are Same where where

Compressor Adiabatic Efficiency Caution - Different pressure ratios are used in this definition Usually Pt exit and Pt inlet (for total-to-total efficiency) Pr = P02/P01 Sometimes Ps exit is used when considering total-to-static efficiency There is no “right” definition of efficiency Only the ideal work is affected by the choice of exit pressure Reference pressure for  

Efficiency Definitions & Relations Compressor & Turbine Adiabatic Efficiency Temperature level effect on Pr Stage Efficiency related to overall Polytropic Efficiency

Why Polytropic Efficiency? Polytropic efficiency: arises in context of a reversible compressor, compressing a gas from an initial state to a final state, but obeying pvn = constant, where n is called the polytropic index Comparison of isentropic  for 2 machines of different pressure ratio (Pr) is not valid, since for equal poly, the compressor with the highest Pr is penalized with a hidden temperature effect  see next chart

Compression System Consider Compression Part of Cycle – Because Constant Pressure Lines Diverge as Entropy Increases, (h1a - h1) + (h1c - h1b) + (h1e - h1d) +(h1g - h1f) > (h2i - h1) so,

Why Polytropic Efficiency? Adiabatic efficiency makes thermodynamic sense for cycle analysis. Changing adiabatic efficiency with varying number of identical stages does not describe fluid mechanics ad is a function of Pr and losses Suppose we combine 2 compressors of equal ad and equal h0 rise to make a compressor of higher pressure rise. The actual h0 =(h0 ), but the isentropic h0 > (h0 )isent,

Polytropic Efficiency - "Small Stage Efficiency" Compare fluid mechanical performance of different machines using poly. Compressor Composed of large number of “small stages” Compressor: Compressor Polytropic Efficiency

Compressor Polytropic Efficiency

Compressor Stage Efficiency Mattingly uses the notation: Each Stage of a Multi-Stage Compressor Has an Adiabatic Efficiency Let psj and tsj Represent Pressure and Temperature Ratio of the jth Stage

Compressor Efficiency From the Stage Efficiency We Have So for N Stages

Compressor Efficiency And the Overall Compressor Efficiency is Where

Compressor Efficiency Only for s = constant Really want s=constant

Compressor Efficiency s can be constant but s is not constant

Compressor Efficiency So for this Special Case (Constant Stage Pressure Ratio and Efficiency)

Example

Example

Compressor Adiabatic Efficiency Actual work can be from temperature or measured rotor torque & mass flow

Turbine Adiabatic (Isentropic) Efficiency

Turbine Adiabatic Efficiency Turbine People Usually Use Expansion Ratio - P1/P2 Consider Cooling Air Later (Just 1st Law Bookkeeping)

Consequences of Molier Diagram Constant Pressure Lines Diverge on Molier Diagram – Looks Slight, but it Matters More Work for given Pr as T increases Less Pr in Latter Stages of Compressor & Turbine Lower overall compressor efficiency as Pr increases Higher overall turbine efficiency as Pr increases.

Polytropic Efficiency - "Small Stage Efficiency" Turbine: Turbine Polytropic Efficiency Turbine Adiabatic Efficiency

Turbine System Now Consider Turbine Part of Cycle – Because Constant Pressure Lines Diverge as Entropy Increases, (h1a - h1) + (h1c - h1b) + (h1e - h1d) +(h1g - h1f) > (h2i - h1) so,

Turbine Efficiency Analysis: Dixon 2.1 Calculate the overall efficiency of turbine ad continued

Turbine Efficiency Analysis: Dixon 2.1 small stage efficiency = poly = 86% overall Pr = P02/P01= r = 4.5 to 1 = 4.5 mean  = 1.333  = 0.6868 and ad-overall = 88.16%

LPT HPT SS-Fan HPC

Component Ideal Actual Figure of Merit Inlet Adiabatic & rev. [isentropic] PR=1, TR=1 Adiabatic, not rev., PR<1, TR=1 PR Compressor Adiabatic & rev. Adiabatic, not rev. Turbine Nozzle

Efficiency Comparison 1 Pr