Theoretical Flammability Diagrams for Oxy-combustion

Abstract

In order to utilize the low-calorific value gases, oxy-combustion technology is generally used to burn the diluted fuel mixture more efficiently, cleanly and safely. Here the thermal balance method is tailored to check the fuel mixture flammability with the help of theoretical flammability diagrams and non-dimensional HQR diagrams. Specifically, the contributions of diluent/oxidant/temperature are checked. Even with limited experimental data, they are found to be a powerful tool to understand the mixture flammability, since they are based on simple principles of energy conservation. This approach provides the foundation for fine-tuning safe operation parameters related to oxy-combustion technologies.

Appendices

Appendix 1: Role of Diluent on the Flammability Envelope of Methane

For any low-calorific value gas mixtures, the diluent (mostly carbon dioxide) is changing the thermal signature of the fuel (combined into a pseudo fuel). For the diluted flammability diagram, a dilution ratio R is defined firstly as

$$ \frac{{x_{D} }}{{x_{F} }} = R $$

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This definition combined with a conservation equation $x_{F} + x_{D} = x_{L}$.

Solve two equations for two variables, we have

$$ x_{D} = \frac{{R \cdot x_{L} }}{1 + R} $$

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$$ x_{F} = \frac{{x_{L} }}{1 + R} $$

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Submitting them into the energy conservation equation at LFL, $x_{F} \cdot Q_{F} + x_{D} \cdot Q_{D} + 1 - x_{L} = x_{F} H_{F}$, we have the thermal balance at critical flammability limits as shown below

$$ \frac{{x_{L} }}{1 + R} \cdot Q_{m} + \frac{{R \cdot x_{L} }}{1 + R} \cdot Q_{D} + \left( {1 - x_{L} } \right) \cdot 1 = \frac{{x_{L} }}{1 + R} \cdot C_{O} H_{O} $$

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$$ \frac{{x_{U} }}{1 + R}Q_{m} + \frac{{R \cdot x_{U} }}{1 + R}Q_{D} + \left( {1 - x_{U} } \right) \cdot 1 = x_{O} \cdot \left( {1 - x_{U} } \right) \cdot H_{O} $$

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where $Q_{m} = Q_{F} \cdot \left( {1 - x_{D} } \right) + Q_{D} \cdot x_{D}$ is the quenching potential of the diluted fuel, $H_{m} = H_{O}$ is unchanged, since the fuel type is the same. The total energy release is scaled down in $C_{m} = C_{O} \cdot \left( {1 - x_{D} } \right)$.

Solve the above equations, we have the flammability envelope bounded by

$$ x_{L} = \frac{1}{{1 + \frac{{C_{m} H_{m} }}{1 + R} - \frac{{Q_{m} }}{1 + R} - \frac{{Q_{D} \cdot R}}{1 + R}}} $$

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$$ x_{U} = \frac{{x_{O} \cdot H_{m} - 1}}{{x_{O} \cdot H_{m} - 1 + \frac{{Q_{m} }}{1 + R} + \frac{{Q_{D} \cdot R}}{1 + R}}} $$

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Forcing $x_{L} = x_{U}$, we have a crossing point ($R_{LU}$, $x_{LU}$) which is the theoretical inertion point.

$$ R_{LU} = \frac{{C_{m} H_{m}^{2} \cdot x_{O} - x_{O} H_{m} Q_{m} - C_{m} H_{m} }}{{x_{O} H_{m} Q_{D} }} $$

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Appendix 2: Role of Oxygen on the Flammability Envelope of Methane

For an oxygen-modified air, the quenching potential of this mixture is defined by

$$ Q_{m} = \left( {1 - x_{O} } \right) \cdot Q_{N} + x_{O} \cdot Q_{O} $$

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where $x_{O}$ is the mole fraction of oxygen in this nitrogen/oxygen mixture. If $x_{O} = 0.2095$, $Q_{m} = 0.992 \times (1 - 0.2095) + 1.046 \times 0.2095 = 1$, which is the normal air used as the reference species.

Following the thermal balance at LFL, we have the controlling LFL equation as

$$ \frac{{x_{L} }}{1 + R} \cdot Q_{F} + \frac{{R \cdot x_{L} }}{1 + R} \cdot Q_{D} + \left( {1 - x_{L} } \right) \cdot Q_{m} = \frac{{x_{L} }}{1 + R} \cdot C_{O} H_{O} $$

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Similarly the governing equation for UFL is

$$ \frac{{x_{U} }}{1 + R}Q_{F} + \frac{{R \cdot x_{U} }}{1 + R}Q_{D} + \left( {1 - x_{U} } \right) \cdot Q_{m} = x_{O} \cdot \left( {1 - x_{U} } \right) \cdot H_{O} $$

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Solve the above 3 equations, we have

$$ x_{L} = \frac{{Q_{m} }}{{Q_{m} + \frac{{C_{O} H_{O} }}{1 + R} - \frac{{Q_{F} }}{1 + R} - \frac{{Q_{D} R}}{1 + R}}} $$

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$$ x_{U} = \frac{{x_{O} \cdot H_{O} - Q_{m} }}{{x_{O} \cdot H_{O} - Q_{m} + \frac{{Q_{F} }}{1 + R} + \frac{{Q_{D} R}}{1 + R}}} $$

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Forcing $x_{L} = x_{U}$, we have the cross point

$$ R_{LU} = \frac{{C_{O} H_{O}^{2} \cdot x_{O} - x_{O} \cdot H_{O} Q_{F} - C_{O} H_{O} Q_{m} }}{{x_{O} \cdot H_{O} Q_{D} }} $$

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Appendix 3: Temperature-Modified Flammability Diagrams

The temperature-dependence of thermal properties is established upon the enthalpy relationships between species provided by NIST chemistry webbook. Reconstruct a simpler correlation for air, we have the enthalpy of air determined by

$$ \begin{gathered} E_{air} = \left( {H_{AFT}^{0} - H_{298.15}^{0} } \right)_{air} = f(T) = 1. 4 8 9 3 {\text{T}}^{ 2} \, + { 29} . 8 6 2 {\text{T }} - { 9} . 3 8 1\hfill \\ \hfill \\ \end{gathered} $$

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Note the input is T/1000, so the coefficients can have more valid digits. Next, define a temperature-dependent enthalpy-scaling factor $\eta (T)$ for measuring the system enthalpy change in reference to air.

$$ \begin{aligned} \eta (T) &= \frac{{E_{i} }}{{E_{air} }} = \frac{{\left( {E_{air}^{1600} - E_{air}^{T} } \right)}}{{\left( {E_{air}^{1600} - E_{air}^{298} } \right)}} = \frac{{42.21 - \left( { 1. 4 8 9 3 {\text{T}}^{ 2} \, + { 29} . 8 6 2 {\text{T }} - { 9} . 3 8 1} \right)}}{{42.21 - \left( { - 0.35} \right)}}\\ &= 1.212 - 0.035 \cdot T^{2} - 0.702 \cdot T \\ \end{aligned} $$

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For the thermal balance at LFL, we have

$$ \left( {\frac{{x_{L} }}{1 + R} \cdot Q_{F} + \frac{{R \cdot x_{L} }}{1 + R} \cdot Q_{D} + \left( {1 - x_{L} } \right)Q_{m} } \right) \cdot \eta = \frac{{x_{L} }}{1 + R} \cdot C_{O} H_{O} $$

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Rearrange the terms, we have a new equation for LFL with R as the only input.

$$ x_{L} = \frac{{\eta Q_{m} }}{{\eta Q_{m} + \left( {\frac{{C_{O} H_{O} }}{1 + R} - \frac{{Q_{F} \cdot \eta }}{1 + R} - \frac{{Q_{D} \cdot R \cdot \eta }}{1 + R}} \right)}} $$

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Similarly, we can solve

$$ \left( {\frac{{x_{U} }}{1 + R}Q_{F} + \frac{{Rx_{U} }}{1 + R}Q_{D} + 1 - x_{U} } \right) \cdot \eta Q_{m} = x_{O} \cdot \left( {1 - x_{U} } \right)H_{O} $$

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Then, we have a new temperature dependent equation for UFL

$$ x_{U} = \frac{{x_{O} \cdot H_{O} - \eta Q_{m} }}{{x_{O} \cdot H_{O} - \eta Q_{m} + \left( {\frac{{Q_{F} }}{1 + R} + \frac{{Q_{D} \cdot R}}{1 + R}} \right) \cdot \eta }} $$

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Forcing$x_{L} = x_{U}$, we have the cross point

$$ R_{LU} = \frac{{C_{O} H_{O}^{2} - \eta Q_{m} H_{O} Q_{F} - C_{O} H_{O} \eta Q_{m} }}{{\eta Q_{m} H_{O} Q_{D} }} $$

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For mixtures without dilution (R = 0), we have the flammability range defined by

$$ x_{L} = \frac{{\eta \cdot Q_{m} }}{{\eta \cdot Q_{m} + C_{O} H_{O} - \eta Q_{F} }} $$

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$$ x_{U} = \frac{{x_{O} \cdot H_{O} - \eta Q_{m} }}{{x_{O} \cdot H_{O} - \eta Q_{m} + \eta Q_{F} }} $$

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Note, the equivalence ratio is defined as

$$ \phi = \frac{{x_{F} }}{{1 - x_{F} }} \cdot \frac{{C_{O} }}{{x_{O} }} $$

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