A system contains 0.15 𝑚3 of a gas at a pressure of 3.8 bar and 150℃ . It is expanded adiabatically till the pressure falls to 1bar. The gas is then heated at a constant pressure till its enthalpy increases by 70 kJ. Determine the total work done. Take Cp = 1 kJ/kg. K and Cv= 0.714 kJ/ kg K.
answer
http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html
Qusn ans
الحل
To determine the total work done, we need to divide the process into two steps:
Step 1: Adiabatic Expansion
Step 2: Constant Pressure Heating
Step 1: Adiabatic Expansion
In an adiabatic process, there is no heat exchange with the surroundings. Hence, the First Law of Thermodynamics simplifies to:
ΔU = -W
where ΔU is the change in internal energy and W is the work done by the system.
To calculate the change in internal energy, we can use the equation:
ΔU = Q - W
where Q is the heat transferred to the system.
As the process is adiabatic, Q = 0, so the equation becomes:
ΔU = -W
The change in internal energy (ΔU) can be expressed as:
ΔU = C_v * m * ΔT
where C_v is the specific heat at constant volume, m is the mass of the gas, and ΔT is the change in temperature.
Since the gas is ideal, we can use the ideal gas equation to calculate the mass of the gas:
PV = mRT
where P is the pressure, V is the volume, R is the specific gas constant, and T is the temperature.
R can be calculated using the equation:
R = Cp - Cv
where Cp is the specific heat at constant pressure.
Plugging in the given values, we can find the initial mass of the gas (m1) using the initial conditions:
P1 = 3.8 bar = 3.8 * 10^5 Pa
V1 = 0.15 m^3
T1 = 150℃ = 150 + 273.15 K
Step 2: Constant Pressure Heating
In a constant pressure process, the work done by the system is given by:
W = PΔV
where P is the pressure and ΔV is the change in volume.
To calculate the change in volume, we can use the equation:
ΔV = Δm * R * ΔT
where Δm is the change in mass of the gas.
The change in mass (Δm) can be found by rearranging the ideal gas equation:
Δm = (P2 * V1 - P1 * V1) / R * T2
Plugging in the given values for the constant pressure heating conditions, we can find the final mass of the gas (m2):
P2 = 1 bar = 1 * 10^5 Pa
T2 = T1 + (ΔH / Cp)
where ΔH is the change in enthalpy.
Finally, we can calculate the total work done by adding the work done in the adiabatic expansion (W1) and the constant pressure heating (W2):
Total work done (W_total) = W1 + W2
Please note that the specific gas constant (R) depends on the particular gas being considered. In this case, the specific gas constant needs to be provided in the question or replaced with the specific gas constant for the given gas.