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TECHNICAL PAPERS

# Defining the Efficiency of a Cooled Turbine

[+] Author and Article Information
J. B. Young

Hopkinson Laboratory, Engineering Department, Cambridge University, Trumpington Street, Cambridge, CB2 1PZ, UKjby@eng.cam.ac.uk

J. H. Horlock

Hopkinson Laboratory, Engineering Department, Cambridge University, Trumpington Street, Cambridge, CB2 1PZ, UKjohn.horlock1@btinternet.com

Sometimes (e.g., Kurzke (4), Timko (6)) the pumping power is added to $Pgross$ in the numerator of Eq. 20 (i.e., $Pgross$ is replaced by the $PX$ of Eq. 3) and a parallel modification is made to the denominator by the addition of an “isentropic” pumping power. We do not follow this practice here.

J. Turbomach 128(4), 658-667 (Mar 14, 2006) (10 pages) doi:10.1115/1.2218890 History: Received August 06, 2005; Revised March 14, 2006

## Abstract

Despite 40 years development of gas turbine cooling technology, there is no general agreement on the most appropriate definition of cooled turbine efficiency; the critical issue is the choice of a hypothetical “ideal” process. This paper reviews the definitions in use and presents new proposals for overcoming the problems. Attention is first focused on a stationary cooled cascade, and it is shown that the commonly used Hartsel efficiency definition (where the gas and coolant streams expand separately in the ideal process) is unsatisfactory. Three “mixed” efficiencies, referred to as the MP (mainstream-pressure), FR (fully reversible), and WP (weighted-pressure) efficiencies are then discussed. The MP ideal process involves mixing of the coolant and the mainstream to give an unchanged mainstream pressure before expansion. This definition, although sometimes used, is unsatisfactory because the efficiency is independent of the coolant supply pressures. The FR and WP efficiencies have not appeared in the literature previously. The FR efficiency is based on a fully reversible ideal process and has the soundest thermodynamic foundation. It is equivalent to a suitably defined rational efficiency and can be directly related to the various cooling losses. However, as it gives a significantly lower value than the Hartsell and MP definitions, it may not appeal to turbine manufacturers. The WP definition is a pragmatic alternative. In the WP ideal process the entropy increase associated with temperature equilibration of the mainstream and coolant flows is allowed in the mixing before expansion. All three mixed efficiencies can be applied to turbine stages with multiple coolant streams. Turbine manufacturers are urged to reconsider their current procedures with a view to standardizing on a thermodynamically sound definition of efficiency.

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## Figures

Figure 1

Schematic diagram of a rotor blade cooling system

Figure 2

Expansion through an uncooled turbine cascade

Figure 3

Expansion through an uncooled turbine stage

Figure 4

Illustrating the notation for Hartsel’s cooled cascade. The mainstream gas, coolant, and mixture have different chemical compositions and therefore different sets of isobars on the (h-s) diagram.

Figure 5

Illustrating Hartsel’s approach for identical perfect gases with p01g=p01c=p0m=p01 and an isentropic mixed expansion. Note the entropy creation during constant pressure mixing.

Figure 6

Hartsel cooled cascade efficiency (εc)HART versus coolant fraction ϕ=mc∕(mg+mc) for gas properties and conditions as given in the text, and (εm)TS=1.0, 0.95, and 0.90 (upper, middle, and lower curves, respectively). The region of interest is ϕ<0.25.

Figure 7

Illustrating the ideal processes in the MP, FR, and WP mixed cascade efficiency definitions. Note that (i) in the (T-s) diagram, only one coolant supply has been indicated; (ii) the dotted line defining point F is straight; and (iii) the WP expansion has been drawn assuming p01c>p01g.

Figure 8

Conceptual fully reversible mixing process (for p01c>p01g) with σT=σP=0. For semi-perfect gases it is easy to show that W1g=W1c, W2g=W2c and Q2g=Q2c.

Figure 9

Conceptual partly reversible mixing process (for p01c>p01g) with σT>0, σP=0. For semi-perfect gases it is easy to show that W2g=W2c and Q2g=Q2c.

Figure 10

Ratio of WP to MP cascade efficiencies versus coolant fraction ϕ=mc∕(mg+mc) for the data given in Table 1 and a single coolant stream supplied at various pressures (0.8<p01c∕p01g<1.2)

Figure 11

Ratio of FR to MP cascade efficiencies versus coolant fraction ϕ=mc∕(mg+mc) for the data given in Table 1 and a single coolant stream supplied at various pressures (0.8<p01c∕p01g<1.2)

Figure 12

Illustrating the notation for Hartsel’s cooled turbine stage. For simplicity, only one coolant supply is shown in the (h-s) diagram.

Figure 13

Illustrating the ideal processes in the MP, FR, and WP mixed stage efficiency definitions. In the (T-s) diagram, only one coolant supply (with p01c>p01g) has been indicated but in practice all coolant streams are included, however minor. Note that the dotted line defining point F is straight.

Figure 14

Three model perfect gas GT cycles with isentropic turbomachinery

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