Question

Air is compressed in a single stage reciprocating compressor at the rate of 1 kg/s from 1 bar at 22°C to 8 bar. If γ = 1.4 and R = 287 J/kgK for air, also take (8)^0.286 = 1.81 and ln(8) = 2.079. Difference in work required in compression was isentropic and isothermal:

1. 240.45 kW
2. 218.32 kW
3. 176.06 kW
4. 64.39 kW

Answer 1

Correct Answer - Option 4 : 64.39 kW

Concept:

Case1: compression follows the isentropic process PV^γ = C

The work done W in kW is given as,

\(W = \frac{γ}{γ-1}P_1V_1[(\frac{P_2}{P_1})^\frac{γ-1}{γ}-1]\)

where, γ is the adiabatic index P₁, P₂, V₁, V₂ are state variables of the initial and final state given by subscripts 1 and 2 respectively.

Case 2: Compression follows an isothermal process PV = C

The work done W in kW is given as,

\(W=P_1V_1ln\frac{V_1}{V_2}=P_1V_1ln\frac{P_2}{P_1}\)

Calculation:

Given: 

P₁ = 1 bar, P₂ = 8 bar, T₁ = 295 K, γ = 1.4

we can find the intial volume V₁ by ideal gas equation, P₁V₁ = mRT₁ 

1× 10⁵ × V₁ = 1 × 287 × 295

V₁ = 0.846 m³

Work done using isentropic process W

 \(W = \frac{1.4}{1.4-1}\times 1\times 10^5\times 0.846[(\frac{8}{1})^\frac{1.4-1}{1.4}-1]\) 

W = 240.269 kW

Work done using isothermal process W

\(W=1\times 10^5 \times 0.846 \times ln\frac{8}{1}\)

W = 175.920 kW

Therefore, the difference between the isentropic and isothermal work is,

W_net = 240.269 - 175.920 kW

W_net = 64.349 kW

All the expressions are arranged to give positive value since we are interested in the magnitude of work input.