Ideal-gas mixtures

Up to now we have treated the working fluids of gas cycles as air and the fuel as a single number, the heat added. Real plants handle mixtures everywhere. The natural gas Northgate buys is mostly methane with some ethane, propane, nitrogen and carbon dioxide. The wastewater plant next door produces biogas, roughly 60 percent methane and 40 percent carbon dioxide, and would like to sell it to Northgate. The gas turbine's exhaust, which the HRSG of Lesson 7 cools, is a mixture of nitrogen, oxygen, carbon dioxide, water vapour and argon. And any plan to capture carbon dioxide from the stack starts from the thermodynamics of separating one gas from a mixture. This lesson develops the tools: how to describe a mixture's composition, how to find its properties from those of its components, and what mixing and separating do to entropy and exergy.

Describing a mixture

A mixture of \(k\) components can be described by mass or by moles. The mass fraction of component \(i\) is \(mf_i = m_i/m_m\) and its mole fraction is \(y_i = N_i/N_m\); each set sums to one. The apparent (or average) molar mass and gas constant of the mixture are

and mass and mole fractions are related by \(mf_i = y_i M_i/M_m\). Gas compositions are usually reported by volume, which for ideal gases is the same as by moles, as the next section shows. Gas analyzers that measure flue gas on a dry basis report the composition after the water vapour has been removed, which matters in combustion calculations (Lesson 12).

Dalton's and Amagat's laws

Two models predict how the components of a mixture share pressure and volume. Dalton's law of additive pressures says that the mixture pressure equals the sum of the pressures each component would exert if it existed alone at the mixture temperature and volume. Amagat's law of additive volumes says that the mixture volume equals the sum of the volumes each component would occupy alone at the mixture temperature and pressure. For ideal gases both are exact, and both lead to the same simple result:

The ratio \(P_i/P_m\) is the pressure fraction and \(P_i\) is the partial pressure of component \(i\); \(V_i/V_m\) is the volume fraction. For real gases at high pressure the two laws give different answers, and neither is exact. A simple fix is Kay's rule, which treats the mixture as a pseudo-pure substance with a pseudocritical temperature and pressure equal to the mole-fraction-weighted averages of the components' critical properties, then reads the compressibility factor from the generalized chart. For the pressures in Northgate's gas supply, a few hundred kilopascals to a few megapascals at ordinary temperatures, the ideal-gas model is adequate.

Properties of ideal-gas mixtures

Because the molecules of ideal gases do not interact, each component behaves as if it alone occupied the mixture's volume at the mixture temperature. Extensive properties therefore add: the mixture's internal energy, enthalpy and entropy are sums of the components' values, each evaluated at the mixture temperature and the component's own partial pressure. Per unit mass or per mole,

Internal energy and enthalpy of an ideal gas depend on temperature only, so partial pressure does not matter for them. Entropy depends on pressure, and here partial pressure is essential:

Northgate's fuel gas and biogas

Tomás applies these ideas to the gas Northgate burns and to the biogas the wastewater plant offers. The program finds the pipeline gas's molar mass, gas constant and mass fractions, its higher heating value per cubic metre, its specific gravity relative to air and its Wobbe index, then repeats the calculation for blends with biogas. It goes on to the gas turbine exhaust and to carbon dioxide separation.

# Ideal-gas mixtures at Northgate: pipeline gas, biogas blends, turbine exhaust and the minimum work of CO2 capture
import math
Ru = 8.314                                          # kJ/kmol K
M = {'CH4': 16.04, 'C2H6': 30.07, 'C3H8': 44.10, 'N2': 28.01, 'CO2': 44.01, 'O2': 32.00, 'H2O': 18.02, 'Ar': 39.95}
HHV = {'CH4': 890.3, 'C2H6': 1559.9, 'C3H8': 2219.9}   # MJ/kmol, products at 25 C with liquid water
gas = {'CH4': 0.950, 'C2H6': 0.025, 'C3H8': 0.003, 'N2': 0.015, 'CO2': 0.007}
biogas = {'CH4': 0.62, 'CO2': 0.37, 'N2': 0.01}

def blend(a, b, frac_b):
    return {s: (1 - frac_b) * a.get(s, 0) + frac_b * b.get(s, 0) for s in set(a) | set(b)}

def props(y):
    Mm = sum(y[s] * M[s] for s in y)
    hhv_mol = sum(y[s] * HHV.get(s, 0) for s in y)
    hhv_vol = hhv_mol / 23.645                        # MJ/m3 at 15 C and 101.325 kPa (23.645 m3/kmol)
    sg = Mm / 28.97
    return Mm, Ru / Mm, hhv_vol, sg, hhv_vol / math.sqrt(sg)

y = gas
Mm, R, hv, sg, wobbe = props(y)
print(f"pipeline gas: M = {Mm:.2f} kg/kmol, R = {R:.4f} kJ/kg K, HHV = {hv:.2f} MJ/m3, SG = {sg:.3f}, Wobbe = {wobbe:.1f} MJ/m3")
print("  mass fractions: " + ", ".join(f"{s} {y[s] * M[s] / Mm:.3f}" for s in y))
for f in (0.1, 0.2, 0.3):
    Mm, R, hv, sg, wobbe = props(blend(gas, biogas, f))
    print(f"  {int(f * 100)} % biogas by volume: HHV = {hv:.2f} MJ/m3, SG = {sg:.3f}, Wobbe = {wobbe:.1f} MJ/m3")

# Gas turbine exhaust (mole fractions) and its specific heat from cubic fits, kJ/kmol K
ex = {'N2': 0.751, 'O2': 0.130, 'CO2': 0.035, 'H2O': 0.075, 'Ar': 0.009}
fit = {'N2': (28.90, -0.1571e-2, 0.8081e-5, -2.873e-9), 'O2': (25.48, 1.520e-2, -0.7155e-5, 1.312e-9),
       'CO2': (22.26, 5.981e-2, -3.501e-5, 7.469e-9), 'H2O': (32.24, 0.1923e-2, 1.055e-5, -3.595e-9), 'Ar': (20.79, 0, 0, 0)}
def cp_mol(s, T): a, b, c, d = fit[s]; return a + b * T + c * T ** 2 + d * T ** 3
Mex = sum(ex[s] * M[s] for s in ex)
for T in (400, 600, 700):
    cp = sum(ex[s] * cp_mol(s, T) for s in ex) / Mex
    print(f"exhaust at {T} K: M = {Mex:.2f}, cp = {cp:.3f} kJ/kg K")
print(f"partial pressure of water in the exhaust at 101 kPa: {0.075 * 101:.1f} kPa")

# Minimum work to separate CO2 completely from a mixture at T0 (per kmol of CO2 captured)
T0 = 298.15
for label, yc in (("air (420 ppm)", 0.00042), ("gas turbine exhaust", 0.035), ("gas boiler flue gas", 0.09), ("cement kiln gas", 0.20)):
    w = -Ru * T0 * (math.log(yc) + (1 - yc) / yc * math.log(1 - yc))
    print(f"  {label:20s} y = {yc:<7}: minimum work = {w / 1000:5.2f} MJ/kmol = {w / M['CO2']:5.0f} kJ/kg CO2")

pipeline gas: M = 16.85 kg/kmol, R = 0.4934 kJ/kg K, HHV = 37.70 MJ/m3, SG = 0.582, Wobbe = 49.4 MJ/m3
  mass fractions: CH4 0.904, C2H6 0.045, C3H8 0.008, N2 0.025, CO2 0.018
  10 % biogas by volume: HHV = 36.27 MJ/m3, SG = 0.615, Wobbe = 46.2 MJ/m3
  20 % biogas by volume: HHV = 34.83 MJ/m3, SG = 0.648, Wobbe = 43.3 MJ/m3
  30 % biogas by volume: HHV = 33.39 MJ/m3, SG = 0.682, Wobbe = 40.4 MJ/m3
exhaust at 400 K: M = 28.45, cp = 1.063 kJ/kg K
exhaust at 600 K: M = 28.45, cp = 1.107 kJ/kg K
exhaust at 700 K: M = 28.45, cp = 1.130 kJ/kg K
partial pressure of water in the exhaust at 101 kPa: 7.6 kPa
  air (420 ppm)        y = 0.00042: minimum work = 21.75 MJ/kmol =   494 kJ/kg CO2
  gas turbine exhaust  y = 0.035  : minimum work = 10.74 MJ/kmol =   244 kJ/kg CO2
  gas boiler flue gas  y = 0.09   : minimum work =  8.33 MJ/kmol =   189 kJ/kg CO2
  cement kiln gas      y = 0.2    : minimum work =  6.20 MJ/kmol =   141 kJ/kg CO2

Whether two gases can be swapped in the same burner depends on the Wobbe index, the higher heating value per unit volume divided by the square root of the specific gravity, \(W = \text{HHV}_v/\sqrt{SG}\). The heat released through a fixed burner orifice at a fixed supply pressure is proportional to it, so gases with similar Wobbe indices give similar heat output and flame behaviour. Northgate's pipeline gas has a Wobbe index of about 49.4 MJ/m³. Adding raw biogas lowers it quickly, to about 46 MJ/m³ at 10 percent and 43 at 20 percent by volume, because the carbon dioxide both dilutes the heating value and makes the gas heavier. Burners are typically adjusted to work within a band of perhaps ±5 percent of the design Wobbe index, so raw biogas blending would be limited to a few percent unless the burners were re-tuned. The usual answer is to upgrade biogas to renewable natural gas by removing most of the carbon dioxide, which brings it to pipeline quality; the separation section below shows the thermodynamic cost.

Gas turbine exhaust

Lesson 7 assumed a specific heat of 1.08 kJ/kg·K for the gas turbine exhaust. The program checks that value with a typical exhaust composition of 75.1 percent nitrogen, 13.0 percent oxygen, 3.5 percent carbon dioxide, 7.5 percent water vapour and 0.9 percent argon, using temperature-dependent molar specific heats for each component. The mixture's molar mass is 28.45 kg/kmol, close to that of air, and its specific heat rises from 1.063 kJ/kg·K at 400 K to 1.107 at 600 K and 1.130 at 700 K. Over the HRSG's temperature range the assumed value is a reasonable average, and a more careful design would integrate the varying specific heat through each section.

The water vapour in the exhaust has a partial pressure of about 7.6 kPa at atmospheric pressure. Its saturation temperature at that pressure, about 40 °C, is the exhaust's dew point: if the HRSG's economizer tubes cool the gas below it, water condenses on them. With natural gas this is harmless, even helpful, since condensing economizers recover the latent heat, but fuels containing sulphur form acids that corrode the tubes, which is why coal and oil boilers keep their stacks well above the acid dew point. Lesson 12 calculates dew points for combustion products in detail.

Entropy of mixing and the work of separation

When two different gases mix at the same temperature and pressure, each expands from its own volume into the whole, and its pressure falls from \(P_m\) to its partial pressure \(y_i P_m\). The entropy generated, per mole of mixture, is

which is always positive because every \(y_i < 1\). Mixing is irreversible, and unmixing, separating a mixture into its components, requires work. The minimum work is that of a reversible separation at the dead-state temperature, \(\bar w_{\min} = T_0 \Delta \bar s_{\text{mix}}\) per mole of mixture. For carbon dioxide capture, the relevant quantity is the minimum work per mole of carbon dioxide removed. If a mixture with mole fraction \(y\) of carbon dioxide is separated completely into pure carbon dioxide and a carbon-dioxide-free remainder,

The result has a clear message for the study. Capturing carbon dioxide from dilute gas turbine exhaust costs more per tonne than capturing it from boiler or industrial flue gas, and capturing it from the air costs more still. Northgate is small for carbon capture, which today is applied mostly at large industrial sources, but the analysis helps the team see why upgrading biogas, which separates carbon dioxide from a 37 percent stream, is relatively cheap, and why it is an established technology.

Mixtures of gases and vapours

Many important mixtures contain a component that can condense. Moist air is the most common: dry air plus water vapour at a partial pressure that cannot exceed the saturation pressure of water at the mixture temperature. Treating both as ideal gases, with the vapour's state fixed by its partial pressure and temperature, is accurate at the low partial pressures involved, and it is the foundation of psychrometrics, the subject of Lesson 11. Combustion products, with their water vapour, are a second example, treated in Lesson 12.

In Canada, natural gas is sold by energy in gigajoules, and its heating value is measured at standard conditions of 15 °C and 101.325 kPa. A gigajoule is about 26.5 m³ of Northgate's pipeline gas.