Formula
\(M = \sum_i y_i M_i \qquad \rho = \frac{p\,M}{R\,T} \qquad \rho_n = \frac{M}{V_m} \qquad c_p = \frac{\sum_i y_i\,C_{p,i}(T)}{M} \qquad h = \int_{T_{ref}}^{T} c_p\,dT \qquad \bar c_p = \frac{h}{T - T_{ref}}\)
Shomate heat capacity per species, \(t = T/1000\):
\(C_{p,i} = A + Bt + Ct^2 + Dt^3 + \frac{E}{t^2}\)
Sutherland viscosity per species and Wilke's mixing rule:
\(\mu_i = \mu_{0,i}\left(\frac{T}{T_{0,i}}\right)^{3/2}\frac{T_{0,i} + S_i}{T + S_i} \qquad \mu = \sum_i \frac{y_i\,\mu_i}{\sum_j y_j\,\Phi_{ij}} \qquad \Phi_{ij} = \frac{\left[1 + (\mu_i/\mu_j)^{1/2}(M_j/M_i)^{1/4}\right]^2}{\sqrt{8\,(1 + M_i/M_j)}}\)
Eucken conductivity per species and the Mason–Saxena form of the Wassiljewa equation (same \(\Phi_{ij}\)):
\(k_i = \mu_i\left(c_{p,i} + \frac{5}{4}\frac{R}{M_i}\right) \qquad k = \sum_i \frac{y_i\,k_i}{\sum_j y_j\,\Phi_{ij}} \qquad \nu = \frac{\mu}{\rho} \qquad Pr = \frac{c_p\,\mu}{k}\)
Variables
| Symbol | Meaning | SI unit |
|---|---|---|
| \(T\) | gas temperature | K |
| \(p\) | absolute pressure | Pa |
| \(y_i\) | volume (mole) fraction of species i: \(y_{O_2}\), \(y_{N_2}\), \(y_{CO_2}\), \(y_{H_2O}\), \(y_{SO_2}\), \(y_{CO}\), \(y_{H_2}\), \(y_{Ar}\) | – |
| \(M_i\), \(M\) | molar mass of species i and of the mixture | kg/kmol |
| \(R\) | universal gas constant, 8 314.46 | J/(kmol·K) |
| \(V_m\) | molar volume at 0 °C and 101.325 kPa, 22.414 | m³/kmol |
| \(\rho_n\) | density at normal conditions | kg/Nm³ |
| \(\rho\) | density at \(T\) and \(p\) | kg/m³ |
| \(C_{p,i}\) | molar heat capacity of species i | J/(mol·K) |
| \(c_p\) | specific heat capacity of the mixture at \(T\) | J/(kg·K) |
| \(\bar c_p\) | mean specific heat between 25 °C and \(T\) | J/(kg·K) |
| \(h\) | enthalpy above 25 °C | J/kg |
| \(\mu_i\), \(\mu\) | dynamic viscosity of species i and of the mixture | Pa·s |
| \(\nu\) | kinematic viscosity | m²/s |
| \(k_i\), \(k\) | thermal conductivity of species i and of the mixture | W/(m·K) |
| \(Pr\) | Prandtl number | – |
| \(T_{ref}\) | reference temperature for enthalpy, 298.15 | K |
Method and standards
The same formulations FireCAD uses in its boiler design software: NIST-JANAF Shomate polynomials for heat capacity (NIST Chemistry WebBook), Sutherland's law for each species' viscosity, Wilke's 1950 rule for the mixture viscosity, the modified Eucken relation for each species' conductivity and the Wassiljewa equation with Mason–Saxena coefficients for the mixture conductivity. Enthalpy is the exact integral of the Shomate heat capacity from 25 °C, the ASME PTC 4 reference temperature.
Assumptions and limits
- Ideal gas (fine for flue gas and air below about 20 bar).
- 250–2 500 K. Water vapour heat capacity below 500 K extrapolates the NIST fit by less than 1 %.
- Transport mixing rules are accurate to about 2–3 %; they are not valid for condensing gas.
References
- NIST-JANAF Shomate heat capacities
- Sutherland viscosity
- Wilke (1950) mixing rule
- Eucken relation
- Mason–Saxena mixing
Questions and answers
Which composition should I enter, wet or dry?
Wet, as the gas actually flows through the heat transfer surfaces and fans. The combustion calculator gives the wet composition; an analyser gives dry, which you must correct for the water vapour.
Why does density fall with temperature but viscosity rise?
Gas density follows the ideal-gas law, ρ ∝ 1/T. Gas viscosity comes from molecular momentum exchange, which increases with molecular speed and therefore with temperature, roughly as T^0.7.
How accurate are the mixing rules?
Wilke's rule for viscosity and the Mason–Saxena form for conductivity are within about 2–3 % of measurements for combustion gases away from condensation. The Shomate heat capacities are NIST fits accurate to better than 0.5 %.