Isentropic Compression of an N₂/CO₂ Ideal-Gas Mixture: 04-BS-10, May 2013, Q6

Question 6: Isentropic Compression of an N₂/CO₂ Ideal-Gas Mixture (15 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Ideal-gas mixture, $y_{N_2}=0.80$, $y_{CO_2}=0.20$ (mole basis). Inlet $P_1=100$ kPa, $T_1=600$ K; exit $P_2=500$ kPa; compression is isentropic (steady flow, adiabatic, reversible).

Find. Compressor work input per unit mass of the mixture.

Approach

Because composition is unchanged across the compressor, the ideal-gas mixing (Gibbs) entropy contribution is identical at inlet and exit and cancels; the mixture's isentropic constraint reduces to a mole-fraction-weighted sum of each pure component's entropy change (evaluated at the same overall pressure ratio, since every species' partial-pressure ratio equals the total pressure ratio). Solve that single equation for the isentropic exit temperature, then get the specific work from the mixture's molar-mass-weighted enthalpy change.

Mixture molar mass. $$M_{mix}=y_{N_2}M_{N_2}+y_{CO_2}M_{CO_2}=0.80(28.013)+0.20(44.01)=\boxed{31.212\ \text{g/mol}}.$$
Isentropic exit temperature from the weighted entropy balance. For an ideal-gas mixture with fixed composition, $\Delta s_{mix}=0$ reduces to $$y_{N_2}\left[s^\circ_{N_2}(T_2)-s^\circ_{N_2}(T_1)\right]+y_{CO_2}\left[s^\circ_{CO_2}(T_2)-s^\circ_{CO_2}(T_1)\right]=R\ln\frac{P_2}{P_1}=8.314\ln(5)=13.381\ \text{J/mol}\cdot\text{K},$$ using each pure component's molar standard-state entropy function $s^\circ_i(T)$ (the mixing-entropy terms are identical on both sides and cancel). Solving numerically for $T_2$: $$\boxed{T_2=881.3\ \text{K}}.$$
Mixture enthalpy change. Ideal-gas enthalpy of each component depends on $T$ only (mixing adds no enthalpy of mixing for an ideal-gas mixture), so $$\Delta h_{mix,molar}=y_{N_2}\left[h_{N_2}(T_2)-h_{N_2}(T_1)\right]+y_{CO_2}\left[h_{CO_2}(T_2)-h_{CO_2}(T_1)\right]=9813.4\ \text{J/mol}.$$
Specific work input. For a steady-flow adiabatic compressor, $w_{in}=h_2-h_1$; converting the molar enthalpy change to a per-unit-mass basis using $M_{mix}$: $$w_{in}=\frac{\Delta h_{mix,molar}}{M_{mix}}=\frac{9813.4\ \text{J/mol}}{31.212\ \text{g/mol}}=\boxed{314.41\ \text{kJ/kg}}.$$

Quantity	Result
$M_{mix}$	31.212 g/mol
$T_2$ (isentropic exit)	881.3 K
$w_{in}$ per kg of mixture	314.41 kJ/kg

Concept & Theory

Topic. Isentropic processes for ideal-gas mixtures of fixed composition.

Key relations. Amagat/Dalton ideal-gas mixture model; component molar entropy $s_i(T,P_i)=s^\circ_i(T)-R\ln(P_i/P_{ref})$; mixture property $=\sum y_i\,(\text{pure-component property})$ for both $h$ and (once mixing terms cancel) $\Delta s$; mixture molar mass $M_{mix}=\sum y_i M_i$.

Why this works. The Gibbs mixing-entropy term for an ideal-gas mixture depends only on composition and total pressure, both of which are unchanged from inlet to exit (only $T$ and total $P$ change, not mole fractions) — so it appears identically on both sides of $\Delta s_{mix}=0$ and cancels exactly, leaving a mole-fraction-weighted sum of the two pure-component entropy functions as the isentropic condition. The same fixed-composition argument makes the mixture enthalpy simply the mole-weighted sum of pure-component enthalpies, with no separate mixing-enthalpy term (true for ideal gases only).

Common pitfall. Trying to use a single "mixture $k$" or "mixture $c_p$" evaluated at only $T_1$ in an isentropic $T_2=T_1(P_2/P_1)^{(k-1)/k}$ shortcut — at 600–900 K, $c_p$ for both N₂ and especially CO₂ varies enough with temperature that this constant-property shortcut noticeably mis-predicts $T_2$; working directly with each component's $s^\circ(T)$ function (or accurate property data) avoids that error.

Source. Cengel & Boles, Thermodynamics: An Engineering Approach, 8th ed., Ch. 13 (Gas Mixtures).

Question 6 of 9: Isentropic Compression of an N₂/CO₂ Ideal-Gas Mixture

Question 6: Isentropic Compression of an N₂/CO₂ Ideal-Gas Mixture (15 marks)

Approach