Question

Carbon dioxide contained in a piston-cylinder arrangement, initially at 6 bar and 400K, undergoes an expansion to a final temperature of 298 k, during which the pressure-volume relationship if pV^1.2 = constant.
Assuming the ideal gas model for the CO2, determine the final pressure, in bar, and the work and heat transfer, each in kJ/kg

Answer 1

Concepts and reason

The gas in the piston cylinder assembly is a closed system. The only work involved is boundary work. The thermodynamic process is a polytropic process following certain relation between pressure and volume. Work done can be determined by finding the area under the pressure-volume diagram.

Fundamentals

A closed system is a quantity of matter surrounded by a fixed or movable boundary. It can only have work and heat transfer with its surroundings, there cannot be any mass transfer with the surroundings.

An ideal thermodynamic process whose deviation from the thermodynamic equilibrium is infinitesimal at any point of time during the process is called a quasistatic process.

Any quasi-equilibrium expansion or compression process can be described by the relation,

$p{V^n} = {\rm{c}}$

Here, pressure is $p$ , volume is $V$ , and constant is $n$ .

A closed system boundary work for a polytropic sprocess can be calculated using the formula,

$W = \frac{{{P_1}{v_1} - {P_2}{v_2}}}{{n - 1}}$

Here, initial volume is ${V_1}$ and final volume is ${V_2}$ .

In a non-cyclic process, the change in the internal energy of a system is equal to net energy added as heat to the system minus the net work done by the system.

Taking $\Delta U$ as a change in internal energy, it is equal to,

$\Delta U = Q - W$ …… (1)

Here, the net heat supplied to the system by its surroundings is Q and the work done by the system is W.

Obtain the properties of carbon dioxide from the properties of various substances table.

$\begin{array}{l}\\R = 0.1889\,{\rm{kJ/kg - K}}\\\\{c_p} = 0.846\,{\rm{kJ/kg - K}}\\\end{array}$

Calculate the final pressure using the polytropic process.

$\begin{array}{l}\\\frac{{{P_2}}}{{{P_1}}} = {\left( {\frac{{{T_2}}}{{{T_1}}}} \right)^{\frac{n}{{n - 1}}}}\\\\\frac{{{P_2}}}{6} = {\left( {\frac{{298}}{{400}}} \right)^{\frac{{1.2}}{{1.2 - 1}}}}\\\\{P_2} = 1.025\,{\rm{bar}}\\\end{array}$

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Calculate the work done during the polytropic process.

$\begin{array}{l}\\W = \frac{{{P_1}{v_1} - {P_2}{v_2}}}{{n - 1}}\\\\W = \frac{{m\left( {R{T_1} - R{T_2}} \right)}}{{n - 1}}\\\\W = \frac{{1 \times 0.1889\left( {400 - 298} \right)}}{{1.2 - 1}}\\\\W = 96.34\,{\rm{kJ/kg}}\\\end{array}$

Calculate the internal energy.

$\begin{array}{l}\\\Delta U = {c_v}\left( {{T_2} - {T_1}} \right)\\\\\Delta U = 0.657\left( {298 - 400} \right)\\\\\Delta U = - 67.014\,{\rm{kJ/kg}}\\\end{array}$

Calculate the heat transfer.

$\begin{array}{l}\\\Delta U = Q - W\\\\Q = \Delta U + W\\\\Q = - 67.014 + 96.34\\\\Q = 29.325\,{\rm{kJ/kg}}\\\end{array}$

Ans:

The final pressure in the piston-cylinder arrangement is $1.025\,{\rm{bar}}$ .

The work done during the polytropic process is $96.34\,{\rm{kJ/kg}}$ .

The heat transfer in the piston-cylinder arrangement is $29.325\,{\rm{kJ/kg}}$ .