Question

2.24 Carbon dioxide (CO2) gas in a piston-cylinder assembly undergoes three processes in series that begin and end at the same state (a cycle). Process 1-2: Expansion from state 1 where p1 = 10 bar, V1 = 1 m3, to state 2 where V2 = 4 m3. During the process, pressure and pV1.5 volume are related by = constant. Process 2-3: Constant volume heating to state 3 where p3 = 10 bar. Process 3-1: Constant pressure compression to state 1. Sketch the processes on p –V coordinates and evaluate the work for each process, in kJ. What is the net work for the cycle, in kJ?

Answer 1

Answer:

Explanation:

Given that:

From process 1 → 2

$P_1 = 10 bar \\ \\ V_1 = 1 m^3 \\ \\ V_2 = 4 m^3$

$PV^{1.5} = \ constant$

$\gamma = 1.5$

Process 2 → 3

The volume is constant i.e $V_2 =V_3 = 4m^3$

$P_3 = 10 \ bar$

Process 3 → 1

P = constant  i.e the compression from state 1

Now, to start with 1 → 2

$P_1V_1^{1.5} = P_2V_2^{1.5}$

$P_2 = P_1 (\dfrac{V_1}{V_2})^{1.5}$

$P_2 = 10 \times (\dfrac{1}{4})^{1.5}$

$P_2 =1.25$

The work-done for the process  1 → 2 through adiabatic expansion is:

$W = \dfrac{1}{1-\gamma}[P_2V_2-P_1V_1]$

We know that 1 bar = $10^5 \ N/m^2$

∴

$W = \dfrac{1}{1-1.5}[1.25 \times 10^5 \times 4- 10 \times 10^5 \times 1]$

$W =1000000 \ J$

$W_{1 \to 2} = 1000 kJ$

For process 2 → 3

Since V is constant

Thus:

W = PΔV = 0

$W_{2 \to 3} = 0$

For process 3 → 1

W = PΔV

$W _{3 \to 1} = P_3(V_1-V_3)$

$W _{3 \to 1} = 10 \times 10^5 (1-4)$

$W _{3 \to 1} = 10 \times 10^5 (-3)$

$W _{3 \to 1} = -3 \times 10^6 \ J$

$W _{3 \to 1} = -3000 \ kJ$

The net work-done now  for the entire system is :

$W_{net} = W_{1 \to 2} + W_{2 \to 3 } + W_{ 3 \to 1 }$

$W_{net} = (1000 + 0 + (-3000)) \ kJ$

$W_{net} =-2000 \ kJ$

The sketch of the processes on p -V coordinates can be found in the image attached below.