4

s s atmospheric pressure and the upper pressure at State 2 represents the pressure after compression of the air by the compressor. The addition of heat between States 2 and 3 occurs in the combustor. Work is then derived from the expansion of the hot combustion gases from States 3 and 4.

In the ideal cycle, the heat input (Q¡n) in the constant-pressure process (Path 2-3) per unit mass is the product of the mass flow rate (ni), the specific heat (cp), and the corresponding change of temperature (T) of the fluid (working fuel) in the process. It is expressed as:

For the constant-pressure heat rejection (Q^) part of the cycle (Path 4-1), the heat output per unit mass is:

Assuming that the specific heats are constant, and neglecting chemical changes that occur during combustion, the ideal thermal efficiency (nBiaywn) can be expressed as:

(Qin - Qout) Cp (T3 - T2)-Cp (T4- T) (T4- Ti) rjD =- = —-L- = 1--

As can be seen from the P-V diagram, p2 = p3 and pi = p4 and the pressure ratio (rp) can be expressed as:

In the ideal cycle, which has no internal losses, the thermal efficiency can also be shown to depend solely on the pressure ratio. Thus, cycle efficiency can also be expressed as:

where k is the ratio of specific heats (cp/cv) at constant pressure and constant volume.

Expressed as a function of volume (V) and compression ratio (rc), the thermal efficiency equation is:

These equations show that the ideal efficiency of the Brayton cycle, which is a constant-pressure combustion cycle with complete expansion, is identical to the ideal efficiency of the Otto cycle, which is a constant-volume combustion cycle with incomplete expansion. Thus, as with the Otto cycle, the theoretical efficiency depends only on the compression ratio or pressure ratio.

In practice, the operation of the gas-turbine engine differs from the ideal Brayton cycle because of irreversibilities in the compressor and turbine, such as friction in the bearings, and pressure drop in the flow passages and combustion chamber. A distinguishing feature of the Brayton cycle is the large amount of compressor work (also called back work) compared to turbine work. The compressor might require 40 to 80% of the output of the turbine. This is considerably higher than the compression work required by reciprocating engines, which are more efficient compressors and expanders. This high-compressor energy requirement is particularly important when actual gas turbine cycle efficiency is considered, because overall efficiency drops rapidly with a decrease in the efficiencies of the compressor and turbine.

The effect of these inefficiencies on actual gas turbine cycle performance can be seen in Figure 10-3. This T-s diagram shows the effect of the irreversible aspects of the real gas turbine cycle. An isentropic compression would attain the Point 2s, whereas the real compressor attains the pressure P2 with an entropy corresponding to Point 2. The turbine expansion lines on the diagram for P2 and P3 illustrate the effect of pressure losses in the combustor and connecting piping. The deviation of the process between Points 4 and 1 from a constant-pressure process illustrates the effect of compressor inlet and turbine exhaust pressure losses on cycle efficiency.

Fig. 10-3 T-s Diagram of Actual Gas Turbine System. Source: Babcock and Wilcox

Accounting for practical application inefficiencies, an equation for actual efficiency becomes:

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