POLYTROPIC EFFICIENCY

It is the isentropic efficiency of one stage of a multistage compressor. This small-stage efficiency is constant for all stages of a compressor with infinite number of stages.

Consider the compression process of a multistage compressor on a T-s diagram as shown in Fig. 15.6 from total pressure p₀₁ to p₀₂ in four stages of equal pressure ratio with intermediate pressure p_0a, p_0b, and p_0c.

Overall isentropic stagnation efficiency of machine is,

Stagnation isentropic efficiency for the stage,

Total actual temperature rise,

Figure 15.6 Concept of polytropic efficiency

Now (ΔT₀) _m/c = (1 − a) + (a − b) + (b − c) + (c − 2)

 

Σ (dT₀)_st = (1 − a) + (a′ − b′′) + (b′ − c′′) + (c′ − 2′′)

On the T-s plot, the constant pressure lines diverge towards the right, therefore,

(a′ − b′′) > (a − b)

(b′ − c′′) > (b − c)

and so on.

Therefore, we can say that

Σ (dT₀)_st > (ΔT₀) _m/c

∴ η_isen (st) > η_isen (m/c)

The small stage efficiency η_isen (st) which is constant for all stages is called polytropic efficiency and is denoted by η_p.

Let the law of compression for the irreversible adiabatic path 1 − 2′ be,

and for the isentropic path 1−2 is,

Eq. (15.20) can be written as:

Differentiating, we get

Differentiating, we get

Eq. (15.21) becomes 

Similarly, for the ideal compression process  we have

From Eq. (15.23), we have 

Integrating between the end states 1 and 2′, we get