Question

Methane gas is compressed at the rate of 3x104 m3 /hr (as measured at 0.1013MPa and 15°C) from 0.69MPa and 26.7°C to 3.45MPa. The compressor operates adiabatically and has an efficiency of 80% compared with isentropic compression. After compression the methane is cooled at a constant pressure of 3.45MPa to a temperature of 38°C. What is the power requirement of the compressor and what is the rate of heat removal in the cooler?

Answer 1

(a). Ws = -1824.99 KW

(b). Q = -148.41 KJ/sec

Step-by-step explanation:

taking a step by step analysis of the problem given we have;

from the data,

Pressure at inlet (P₁) = 0.69 MPa = 0.69 ˣ 10³ KPa

Pressure at outlet (P₂) = 3.45 MPa = 3.45 ˣ 10³ KPa

T₁ = 26.7 ⁰C = 299.7 K

T₂ = 28 ⁰C = 311 K

Also the flow rate of methane gas (Q) = 3 ˣ 10⁴ m³/hr

density of the methane gas (Z) = 0.717 Kg/m³

we have that the mass of flow rate of methane (м) =( 3 ˣ 10⁴ × 0.717) / 3600

м = 5.975 Kg/sec

Also the no of moles of methane used (n) = 5.975/16 = 0.3734

The question ask us to calculate the power requirement of the compressor.

Given;

Ws = V/V-1 P₁V₁[1 - [P₂/P₁]∧(V-1/V)] ------------ (1)

∴ where V = adiabatic index for methane = 1.313

from P₁V₁ = nRT₁

V₁ = (0.3734 ˣ 8.314 ˣ 299.7) / 0.69 ˣ 10³ = 1.3484 m³

V₁ = 1.3484 m³.

inputting value of V₁ into equation (1) we have

Ws = 1.313/1.313-1 (0.69 ˣ 10³ ˣ 1.3484)[1 - [3.45ˣ10³ / 0.69 ˣ 10³ ]∧(1.313-1/1.313)]

Ws = -1824.99 kW

where the negative sign rep that work is required i.e. work is done on system for compression.

(ii). the other question asks us the rate of heat removal in the cooler.

but first, we have that the efficiency of compressor is given thus;

É = ΔHs / ΔH = 2281.24 kJ/sec

where ΔHs rep the isentropic compression.

Now the heat removal becomes;

Q = MCpΔT

here Cp = 2.20 kJ/KGK

Q = MCp(T₁-T₂) = 5.97 ˣ 2.20 ˣ (299.7 - 311) = -148.414 KJ/sec

Q = -148.414 KJ/sec

cheers i hope this helped !!!