Work and efficiencies in turbine stages

Energy equation in various forms and the concepts discussed for nozzles can employed for turbines

In turbines the difference is on account of the presence of shaft work

Ideal and actual expansion process in turbine

In the isentropic process (1-2s), on account of work transfer there is a drop in stagnation enthalpy and the entropy remains same on account of isentropic process

The actual expansion process (irreversible adiabatic) 1-2, on account of irreversibility there is an increase in entropy

Stagnation pressures at exit cannot be compared with initial stagnation pressures because of the work transfer.

The actual work at the turbine can be determined from the change in stagnation enthalpies at the entry and exit.

For perfect gas 01 02aw h h

Total to total Efficiency Efficiency of turbine is ratio of actual to ideal work for the

same Pressure ratio (pr=p1/p2) In turbines the actual work can be measured and the ideal work

is hypothetical and depends on the manner it is defined If the ideal work is defined as the work obtained during the

isentropic expansion from the stagnation state O1 to O2s , then the efficiency based on this is known as the total to total efficiency

Here the KE of the gas at the exit is not considered as wasted since it is contained in the term h02s ,

H02s = h2s + c2s2/2

att

s

ww

01 02

01 02tt

s

h hh h

01 02

01 02tt

s

T TT T

(8)

The stagnation pressure lines for p02s and p02 are different. However the distance between them is small.

The stagnation pressure ratio is

This expression when substituted in eqn. 8

For given stagnation temperature, pressure ratio and efficiency the output power at the shaft is

Total to static efficiency

Some turbines the KE of the out going jet is lost because it is not used after the turbineeg. Some turbine stages exhaust in to atmosphere or in a closed space like condenser

In such case the ideal work is the work done between the states O1 and 2s

The actual work remains the same as the before The total to static efficiency is given by

If

Finite stage efficiency A stage with finite pressure drop is a finite turbine stage. In multi-stage turbines along with overall efficiency, the

efficiencies of individual stages are also important Different stages with same pressure ratio located in different

regions in the h-s plane will give different values of work output.

For a steady flow processdw = -v dp

This implies that for the same pressure drop more work will be done with higher values of v

At each stage the work done is proportional to the initial temperature of the gas

Effect of reheat

Total expansion process 1-2 is divided in to four stages of the same efficiency (ηst) and pressure ratio

Consider the overall efficiency of the expansion is ηT

Actual work during the expansion process 1-2 is wa = ηT ws

If the isentropic or ideal work in the stages are ws1, ws2 , ws3 and ws4

The constant pressure lines in a h-s plane must diverge towards right, therefore

This makes the overall efficiency of the turbine greater than the individual stage efficiency.

ηT > ηst

The quantity ∑ ∆ws / ws is known as the Reheat factor This factor is always greater than unity The effect depicted by ηT > ηst , is due to a thermodynamic

effect called “ Reheat ” This does not imply any heat transfer to the stages from

outside It is the reappearance of stage losses as increased enthalpy

during the constant pressure heating (reheating) processes AX, BY, CZ, D2

Infinitesimal stage efficiency

Expanding the binomial expression on RHS and ignoring the terms beyond the second

This differential equation is valid along the actual expansion process. On integration eqn. 9 yields,

This relation defines the actual expansion line in a finite stage or a multistage machine between two given states

Here the value of infinitesimal or small stage efficiency (ηp) is constant

(10)

(9)

The value of ηp must be determined to use the above eqn. 10 for a given expansion between two states.

Integrating eqn.9 between the given two states 1 and 2,

(11)

Irreversible adiabatic expansion process (the actual expansion process) can be considered as equivalent to polytropic process

So eqn.11 can be written as

Equating the indices we have,

When ηp = 1, n = γ. The actual expansion line coincides with the isentropic expansion and the above equations will be valid for an isentropic process

The efficiency of a finite stage can be expressed in terms of the small stage efficiency.

Variation of stage efficiencies with pressure ratio at constant ηp