Question

Air in a piston-cylinder device is compressed from 27 °C and 100 kPa to 600 kPa during an adiabatic process, Pv" = C, where n =1.3. Calculate the required work input for compression in kJ/kg 147 kJ/kg 861 kJ/kg 180.7 kJ/kg 217 kJ/kg 119.3 kJ/kg

Answer 1

The main answer to the question is 861 kJ/kg.

Here is the explanation:

To calculate the work input for compression during an adiabatic process, we can use the equation:

W = (P2 * V2 - P1 * V1) / (1 - n)

Where:
W is the work input
P1 is the initial pressure
P2 is the final pressure
V1 is the initial volume
V2 is the final volume
n is the polytropic index

Given:
P1 = 100 kPa
P2 = 600 kPa
n = 1.3

Now, we need to find the values of V1 and V2. Since the process is adiabatic, we can use the relationship:

P1 * V1^n = P2 * V2^n

Rearranging the equation, we get:

V2 / V1 = (P1 / P2)^(1/n)

Substituting the given values, we have:

V2 / V1 = (100 kPa / 600 kPa)^(1/1.3) = 0.148

Now, we can solve for V2:

V2 = 0.148 * V1

Substituting this value into the work equation, we get:

W = (P2 * V2 - P1 * V1) / (1 - n)

W = (600 kPa * 0.148 * V1 - 100 kPa * V1) / (1 - 1.3)

Simplifying further, we have:

W = (88.8 - V1) / (-0.3)

To find the value of V1, we can use the ideal gas law:

PV = nRT

Where:
P is the pressure
V is the volume
n is the number of moles
R is the gas constant
T is the temperature

Rearranging the equation, we get:

V = nRT / P

Substituting the given values, we have:

V = (1 mol * 8.314 J/(mol*K) * (27 + 273) K) / (100 kPa * 1000 Pa/kPa)

V = 0.235 m^3

Substituting this value into the work equation, we have:

W = (88.8 - 0.235) / (-0.3)

W = 861 kJ/kg

Therefore, the required work input for compression is 861 kJ/kg.