Work and heat transfer calculation in a closed system
Find the pressure, temperature and volume at the end of each of the four processes. Then calculate the work done during each process and the net amount of work done. Finally calculate the net amount of heat transfer comparing the very initial and the very final states.
Nitrogen T1 (°C): 90, P1 (kPa): 530, m(kg): 9.3
process1: Isobaric expansion and the volume increased by 70%
process2: Isothermal expansion to a pressure of 10% of the initial pressure.
process3: Polytrpic comprission with n = 1.2 to the same initial process.
process4: Isochoric cooling to reach a pressure equal to 40% of the initial pressure
To calculate the pressure, temperature, and volume at the end of each process, we can use 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 ideal gas constant, and T is the temperature.
Given the initial conditions:
T1 = 90°C = 90 + 273.15 = 363.15 K (temperature)
P1 = 530 kPa (pressure)
m = 9.3 kg (mass)
Process 1: Isobaric Expansion
In this process, the volume increases by 70% while the pressure remains constant. To calculate the final volume (V2), we can use the equation:
V2 = V1 + (70% * V1) = V1 * 1.7
where V1 is the initial volume.
We don't have information about the number of moles, so we need to calculate it using the mass and molar mass of nitrogen (28.0134 g/mol).
n = m / M
where M is the molar mass.
Now we can use the ideal gas law to find the final pressure (P2) and temperature (T2):
P2 = (P1 * V1) / V2
T2 = (P2 * V2) / (n * R)
Process 2: Isothermal Expansion
In this process, the temperature remains constant while the pressure decreases to 10% of the initial pressure (0.1 * P1). The final pressure (P3) is given.
The final volume (V3) can be calculated by rearranging the ideal gas law equation:
V3 = (P1 * V2) / P3
To find the final temperature (T3), we use the given final pressure (P3) and calculate it using the ideal gas law:
T3 = (P3 * V3) / (n * R)
Process 3: Polytropic Compression
In this process, a polytropic compression occurs with n = 1.2 (specific heat ratio). The initial and final volume (V3 and V4) are the same.
To find the final pressure (P4), we can use the equation for a polytropic process:
P4 / P3 = (V3 / V4)^(n-1)
To find the final temperature (T4), we can use the ideal gas law:
T4 = (P4 * V4) / (n * R)
Process 4: Isochoric Cooling
In this process, the volume remains constant, and the pressure decreases to 40% of the initial pressure (0.4 * P1). The final pressure (P5) is given.
To find the final temperature (T5), we use the given final pressure (P5) and calculate it using the ideal gas law:
T5 = (P5 * V4) / (n * R)
Now that we have the pressure, temperature, and volume at the end of each process, we can calculate the work done during each process using the formulas for different types of work (isobaric, isothermal, polytropic, and isochoric).
Work (W) = P * ΔV
where P is the pressure and ΔV is the change in volume.
Finally, to find the net amount of work done, we can simply sum up the work done during each process.
To calculate the net amount of heat transfer, we need to compare the initial and final states. It is given that process 1 starts from the initial state and process 4 ends in the final state. We assume the system is adiabatic, meaning there is no heat transfer during the processes.
You can now use these equations and calculations to determine the pressure, temperature, volume, work, and net amount of work for each process.