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A two-stage reciprocating compressor is used to provide compressed air for the pneumatic control system of a concrete mixing plant, with a mass flow rate of 8.0 kg/min. The air is at 100.2 kPa during the suction process and delivered to 1.2 MPa. The compression process follows the equation [tex]PV^{1.2} = C[/tex], with ideal intermediate pressure and perfect intercooling. Assume air density is [tex]1.1 \, \text{kg/m}^3[/tex]. Water has a [tex]C_p = 4.186 \, \text{kJ/(kg-K)}[/tex].

What is the intermediate pressure, [tex]P_x[/tex], in kPa?

Answer :

The intermediate pressure, Px, in the two-stage reciprocating compressor is approximately 11.4 kPa.

The intermediate pressure, Px, in the two-stage reciprocating compressor can be determined using the ideal gas law equation, PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature.

First, let's convert the mass flow rate of 8.0 kg/min to the number of moles. We can use the equation n = m/M, where n is the number of moles, m is the mass, and M is the molar mass. The molar mass of air is approximately 28.97 g/mol.

So, n = (8.0 kg/min * 1000 g/kg) / 28.97 g/mol = 276.3 mol/min.

Next, we need to determine the volume. We can use the equation V = m/d, where V is the volume, m is the mass, and d is the density. The density of air is given as 1.1 kg/m3.

So, V = (8.0 kg/min * 1000 g/kg) / (1.1 kg/m3) = 7272.7 m3/min.

Now, let's calculate the initial pressure, P1, and temperature, T1, using the equation PV = nRT.

P1 * V = n * R * T1

(100.2 kPa) * (7272.7 m3/min) = (276.3 mol/min) * (8.314 J/(mol·K)) * T1

Simplifying the equation, we can solve for T1:

T1 = (100.2 kPa * 7272.7 m3/min) / (276.3 mol/min * 8.314 J/(mol·K)) = 316.4 K

Now, we can determine the final pressure, P2, using the equation PV = nRT, where V is the volume of the compressed air.

(Px) * (V) = (276.3 mol/min) * (8.314 J/(mol·K)) * (T1)

Simplifying the equation, we can solve for Px:

Px = (276.3 mol/min * 8.314 J/(mol·K) * T1) / V

Given that V is equal to (7272.7 m3/min), we can substitute the values and solve for Px.

Px = (276.3 mol/min * 8.314 J/(mol·K) * 316.4 K) / (7272.7 m3/min)

After performing the calculation, the intermediate pressure, Px, is found to be approximately 11.4 kPa.

In summary, the intermediate pressure, Px, in the two-stage reciprocating compressor is approximately 11.4 kPa.

To learn ideal gas equation here:

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