Consider a cylinder containing air at 1200 kPa and 350 C and then, the air is expanded
to 140 kPa with a reversible adiabatic process. Calculate the speci�c work (kJ=Kg)
done by the gas. Assume calorically perfect gas
To calculate the specific work done by the gas during the process, we need to use the First Law of Thermodynamics, which states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.
In this case, since the process is adiabatic (no heat transfer), the work done is solely due to changes in internal energy. For a calorically perfect gas, the specific internal energy is given by the equation:
u2 - u1 = -(P2 - P1) / (γ - 1)
where u1 and u2 are the initial and final specific internal energies respectively, P1 and P2 are the initial and final pressures, and γ is the specific heat ratio.
First, let's find the initial specific internal energy (u1). To do that, we'll use the equation:
u1 = c_v * T1
where c_v is the specific heat at constant volume and T1 is the initial temperature.
Since you mentioned that the gas is air, we need to know the specific heat ratio (γ) and the specific heat at constant volume (c_v) for air at constant pressure. For air, γ is approximately 1.4 and the specific heat at constant volume (c_v) is about 0.718 kJ/(kg·K).
Given:
P1 = 1200 kPa
T1 = 350 °C
We need to convert the temperature from Celsius to Kelvin to use it in the equation.
T1 = 350 + 273.15 = 623.15 K
Now we can calculate the initial specific internal energy (u1):
u1 = c_v * T1 = 0.718 kJ/(kg·K) * 623.15 K = 447.91 kJ/kg
Next, we need to find the final specific internal energy (u2). To do that, we can rearrange the equation mentioned earlier:
u2 = u1 - (P2 - P1) / (γ - 1)
Given:
P2 = 140 kPa
Again, we need to convert the pressure from kPa to Pa to use it in the equation.
P2 = 140 kPa = 140,000 Pa
Now we can calculate the final specific internal energy (u2):
u2 = 447.91 kJ/kg - (140,000 Pa - 1200,000 Pa) / (1.4 - 1) = 298.58 kJ/kg
Finally, we can calculate the specific work done by the gas using the equation:
W = u2 - u1
W = 298.58 kJ/kg - 447.91 kJ/kg = -149.33 kJ/kg
The specific work done by the gas is -149.33 kJ/kg. Since the value is negative, it indicates work done on the gas rather than work done by the gas.