Interaction of a H₂O/Ar Plasma Jet with Nitrogen Atmosphere: Effect of the Method for Calculating Thermophysical Properties of the Gas Mixture on the Flow Field

Abstract

The effect of calculating thermodynamic and transport properties of a gas mixture with mixing rules on the flow field in the modeling of a thermal plasma jet was studied. A 3D large eddy simulation model of a non-transferred direct current hybrid water/argon plasma torch issuing in nitrogen atmosphere at 400 K was developed to compare three different models for the calculation of transport and thermodynamic properties of the ternary gas mixture. In the first model, thermodynamic and transport properties of the pure gases are used with mixing rules to estimate the mixture properties. In the second model, the properties of plasma gas (Ar/H₂O) are calculated rigorously and mixing rules are used for estimating the properties of the mixture of plasma gas and nitrogen. In the third model, the thermodynamic and transport properties of the ternary gas mixture are calculated rigorously without any mixing rules. From numerical results, the error introduced by using mixing rules was evaluated through comparison of calculated temperature, velocity and concentration profiles of the flow field at different positions downstream of the torch exit nozzle. It was found that the differences in transport properties between the exact solutions and the results from calculation with mixing rules can yield significantly different flow fields.

Appendix 1: Expression for the Diffusion Flux in a Three-Component Mixture in Terms of Binary Combined Diffusion Coefficients

A gas mixture of Ar (A), H₂O (B) and N₂ (C) is considered. The diffusion flux of gas A (consisting of species 1 to p) can be written as:

$$\bar{J}_{A} = \frac{{n^{2} }}{\rho }\sum\limits_{i = 1}^{p} {k_{i} m_{i} \sum\limits_{j = 1}^{q} {m_{j} D_{ij} \nabla x_{j}^{(3)} } } - \sum\limits_{i = 1}^{p} {k_{i} } D_{i}^{T} \nabla \ln T$$

(8)

in which i = 1,…,q corresponds to the species making up gas B and C. x ⁽³⁾ _j represents the mole fraction of species j in the three-component gas mixture. For further details on the derivation of this expression, see Murphy [38].

Because the mole fraction of species j (x ⁽³⁾ _j ) is function of temperature and composition, and because composition is determined by the mole fractions of two out of three gases, the gradient of the mole fraction becomes:

$$\nabla x_{j}^{(3)} = \left. {\frac{{\partial x_{j}^{(3)} }}{{\partial x_{B} }}} \right|_{{x_{C} ,T}} \nabla x_{B} + \left. {\frac{{\partial x_{j}^{(3)} }}{{\partial x_{C} }}} \right|_{{x_{B} ,T}} \nabla x_{C} + \left. {\frac{{\partial x_{j}^{(3)} }}{\partial T}} \right|_{{x_{B} ,x_{C} }} \nabla T$$

(9)

in which x _B and x _C are respectively the mole fractions of gas B and gas C at temperature T. The species 1 to q are the collection of species originating from both gas B and gas C. Since no combination species are considered between species from B and species from C, the species 1,…,q can be split up into 1,…,q _B (species from gas B) and q _B  + 1,…,q _C (species from gas C). The following relations are then being considered:

$$\left. {{\text{For}}\;j = 1, \ldots ,q_{B} :\frac{{\partial x_{j}^{(3)} }}{{\partial x_{C} }}} \right|_{{x_{B} ,T}} \cong 0\;{\text{and}}\;{\text{for}}\;j = q_{B} + 1, \ldots ,q_{C} :\left. {\frac{{\partial x_{j}^{(3)} }}{{\partial x_{B} }}} \right|_{{x_{C} ,T}} \cong 0$$

(10)

For j = 1,…,q _B :

$$\left. {\frac{{\partial x_{j}^{(3)} }}{\partial T}} \right|_{{x_{B} ,x_{C} }} \cong \left. {\frac{{\partial x_{j}^{(2)} }}{\partial T}} \right|_{{x_{B} }}$$

(11)

$$\left. {\frac{{\partial x_{j}^{(3)} }}{{\partial x_{B} }}} \right|_{{x_{C} ,T}} \cong \left. {\frac{{\partial x_{j}^{(2)} }}{{\partial x_{B} }}} \right|_{T}$$

(12)

with x ⁽²⁾ _j the mole fraction of species j in the binary gas mixture of A and B with the same mole fraction of gas B, x _B as in the three-component gas mixture.For j = q _B  + 1,…,q _C :

$$\left. {\frac{{\partial x_{j}^{(3)} }}{\partial T}} \right|_{{x_{B} ,x_{C} }} \cong \left. {\frac{{\partial x_{j}^{(2)} }}{\partial T}} \right|_{{x_{C} }}$$

(13)

$$\left. {\frac{{\partial x_{j}^{(3)} }}{{\partial x_{C} }}} \right|_{{x_{B} ,T}} \cong \left. {\frac{{\partial x_{j}^{(2)} }}{{\partial x_{C} }}} \right|_{T}$$

(14)

with x ⁽²⁾ _j the mole fraction of species j in the binary gas mixture of A and C with the same mole fraction of gas C, x _C as in the three-component gas mixture.

With the definitions of combined diffusion coefficients:

$$\bar{D}_{AB}^{x} = \frac{1}{{m_{A} m_{B} }}\sum\limits_{i = 1}^{p} {k_{i} m_{i} \sum\limits_{j = 1}^{{q_{B} }} {m_{j} D_{ij} \left. {\frac{{\partial x_{j}^{(2)} }}{{\partial \bar{x}_{B} }}} \right|} }_{T}$$

(15)

$$\bar{D}_{AC}^{x} = \frac{1}{{m_{A} m_{C} }}\sum\limits_{i = 1}^{p} {k_{i} m_{i} \sum\limits_{{j = q_{B} + 1}}^{{q_{C} }} {m_{j} D_{ij} \left. {\frac{{\partial x_{j}^{(2)} }}{{\partial \bar{x}_{C} }}} \right|} }_{T}$$

(16)

$$\bar{D}_{AB}^{T} = - \frac{{n^{2} }}{\rho }\sum\limits_{i = 1}^{p} {k_{i} m_{i} \sum\limits_{j = 1}^{{q_{B} }} {m_{j} D_{ij} \left. {\frac{{\partial x_{j}^{(2)} }}{\partial T}} \right|} }_{{x_{B} }} + \sum\limits_{i = 1}^{p} {k_{i} D_{i}^{{T\left( {AB} \right)}} }$$

(17)

with D ^T(AB) _i the thermal diffusion coefficient of species i in the binary mixture of A and B

$$\bar{D}_{AC}^{T} = - \frac{{n^{2} }}{\rho }\sum\limits_{i = 1}^{p} {k_{i} m_{i} \sum\limits_{{j = q_{B} + 1}}^{{q_{C} }} {m_{j} D_{ij} \left. {\frac{{\partial x_{j}^{(2)} }}{\partial T}} \right|} }_{{x_{C} }} + \sum\limits_{i = 1}^{p} {k_{i} D_{i}^{{T\left( {AC} \right)}} }$$

(18)

with D ^T(AC) _i the thermal diffusion coefficient of species i in the binary mixture of A and C

The diffusion flux of gas A becomes:

$$\begin{aligned} \bar{J}_{A} &= \frac{{n^{2} }}{\rho }\left( {m_{A} m_{B} \bar{D}_{AB}^{x} (x_{B} ,T)\nabla x_{B} + m_{A} m{}_{C}\bar{D}_{AC}^{x} (x_{C} ,T)\nabla x_{C} } \right) \\ & \quad - \bar{D}_{AB}^{T} (x_{B} ,T)\nabla \ln T - \bar{D}_{AC}^{T} (x_{C} ,T)\nabla \ln T + S \\ \end{aligned}$$

(19)

with the extra term S:

$$S = \sum\limits_{i = 1}^{p} {k_{i} D_{i}^{{T\left( {AB} \right)}} \nabla \ln T} + \sum\limits_{i = 1}^{p} {k_{i} D_{i}^{{T\left( {AC} \right)}} \nabla \ln T} - \sum\limits_{i}^{p} {k_{i} D_{i}^{T\left( 3 \right)} \nabla \ln T}$$

(20)

in which D ^T(3) _i is the thermal diffusion coefficient of species i in the three-component mixture. The term S consists of the terms describing the Soret effect, which is considered small, therefore the term S is neglected in the expression for the diffusion flux of gas A in the three-component mixture of A, B and C.