Modified multi-component gas transport formulation with phoretic effects

Abstract

The understanding of multicomponent mass transport processes is essential for modeling and optimization of many systems, such as fuel cells. The understanding of individual species behavior becomes quite significant at micro-nano scales, where average mixture models are not very accurate. Also at micro-nano scales, additional phoretic transport is present due to strong local temperature and pressure gradients, which is discussed by Chakraborty and Durst (Phys Fluids 19(8):088104-01–088104-04, 2007). To account for multicomponent mass transport, recently proposed mass transport model by Kerkhof and Geboers (AIChE J 51(1):79–121, 2005), provides a way to look into individual components. This article presents extended multicomponent mass transport equations for micro-nano scales within continuum region. The Kerkhof–Geboers theory and the modifications suggested by Chakraborty and Durst (2007) have been combined together to form a new set of equations. An extensive order of magnitude analysis has been done on the modified equations. The application of new equations to different problem situations has also been discussed. It is shown that at very small length scales and for highly diffusive transport, the phoretic transport dominates the system, thus rendering the conventional equations erroneous.

Appendices

Appendix A: Mass and momentum conservation equation

1.1 Mass conservation equation

The mass conservation equation given in Eq. 13 can be expanded by using ideal gas law ($\rho_i\propto {\frac{p_i} {T}}$) and rearranged to get the following equation for mass conservation:

$$ 0=\frac{p_i} {T}\left[\frac{\partial v_{x,i}} {\partial x}+\frac{\partial v_{z,i}} {\partial z}\right]+\frac{1} {T}\left[v_{x,i}\frac{\partial p_i} {\partial x}+v_{z,i}\frac{\partial p_i}{\partial z}\right]-\frac{p_i} {T^2}\left[v_{x,i}\frac{\partial T} {\partial x}+v_{z,i}\frac{\partial T}{\partial z}\right]+\frac{1}{T}\left[\left(\frac{D_i}{T}\frac{\partial T} {\partial x}-\frac{D_i}{p_i}\frac{\partial p_i} {\partial x}\right)\frac{\partial p_i}{\partial x}+\left(\frac{D_i} {T}\frac{\partial T}{\partial z}-\frac{D_i}{p_i}\frac{\partial p_i} {\partial z}\right)\frac{\partial p_i}{\partial z}\right]-\frac{p_i}{T^2}\left[\left(\frac{D_i}{T}\frac{\partial T}{\partial x}-\frac{D_i}{p_i}\frac{\partial p_i} {\partial x}\right)\frac{\partial T}{\partial x}+\left(\frac{D_i}{T}\frac{\partial T} {\partial z}-\frac{D_i}{p_i}\frac{\partial p_i}{\partial z}\right)\frac{\partial T}{\partial z}\right]+\frac{p_i}{T}\left[\frac{D_i}{T}\frac{\partial^2T} {\partial x^2}-\frac{D_i}{p_i}\frac{\partial^2p_i}{\partial x^2}-\frac{D_i} {T^2}\left(\frac{\partial T}{\partial x}\right)^2+\frac{D_i}{p_i^2}\left(\frac{\partial p_i} {\partial x}\right)^2+\frac{D_i}{T}\frac{\partial^2T} {\partial z^2}-\frac{D_i}{p_i}\frac{\partial^2p_i}{\partial z^2}-\frac{D_i} {T^2}\left(\frac{\partial T}{\partial z}\right)^2+\frac{D_i}{p_i^2}\left(\frac{\partial p_i} {\partial z}\right)^2\right]$$

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The expanded equation consists of the conventional continuity equation given in Eq. 4, and additional phoretic terms. This equation can be non-dimensionalized using the scaling parameters described in Sect. 3, to get the following equation:

$$ 0={\frac{\rho U^2}{2T_o}}\,{\frac{p_i^{\ast}}{T^{\ast}}}\left[{\frac{U}{L}}\underset{\approx O(10^{-3})}{\underbrace{\left[{\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}+{\frac{\partial v_{z,i}^{\ast}}{\partial z^{\ast}}}\right]}}+{\frac{D_i}{L^2}}\underset{\approx O(1)}{\underbrace{\left[{\frac{1}{T^{\ast}}}\,{\frac{\partial^2T^{\ast}} {\partial x^{*2}}}-{\frac{1} {p^{\ast}_i}}\,{\frac{\partial^2p_i^{\ast}} {\partial x^{*2}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^2\right]}}+{\frac{D_i} {L^2}}\underset{\approx O(10^{-6})}{\underbrace{\left[{\frac{1} {T^{\ast}}}\,{\frac{\partial^2T^{\ast}} {\partial z^{*2}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial^2p^{\ast}} {\partial z^{*2}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial z^{\ast}}}\right)^2+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)^2\right]}}\right]+{\frac{D_i\rho U^2} {2L^2T_0}}\,{\frac{1} {T^{\ast}}}\left[\underset{\approx O(1)}{\underbrace{{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)}}+\underset{\approx O(10^{-6})}{\underbrace{{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)}}\right]+{\frac{\rho U^3}{2T_0L}}\left[\underset{\approx O(10^{-3})}{\underbrace{{\frac{1}{T^{\ast}}}\left(v_{x,i}^{\ast}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}+v_{z,i}^{\ast}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)}}-\underset{\approx O(10^{-3})}{\underbrace{{\frac{p_i^{\ast}} {T^{*2}}}\left(v_{x,i}^{\ast}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}+v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)}}\right]-{\frac{D_i\rho U^2}{2T_0L^2}}\left[\underset{\approx O(1)}{\underbrace{{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)}}+\underset{\approx O(10^{-6})}{\underbrace{{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)}}\right] $$

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Finally, after removing the negligible terms and again rearranging the equation, we get the continuity equation as follows:

$$ 0={\frac{p_i^{\ast}}{T^{\ast}}}\left[{\frac{\partial v_{x,i}^{\ast}} {\partial x^{\ast}}}+{\frac{\partial v_{z,i}^{\ast}}{\partial z^{\ast}}}\right]+{\frac{1}{T^{\ast}}}\left(v_{x,i}^{\ast}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}+v_{z,i}^{\ast}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)-{\frac{p_i^{\ast}}{T^{*2}}}\left(v_{x,i}^{\ast}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}+v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)+{\frac{1}{Pe_i}}\,{\frac{p_i^{\ast}}{T^{\ast}}}\left[{\frac{1} {T^{\ast}}}\,{\frac{\partial^2T^{\ast}}{\partial x^{*2}}}-{\frac{1} {p^{\ast}_i}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{*2}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)^2\right]+{\frac{1}{Pe_i}}\left[{\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right]\left[{\frac{1}{T^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}-{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\right] $$

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Rearranging the terms and using ideal gas law again, we get the final continuity equation as follows, where the new variable $\rho_i^{\ast}$ is defined as $\rho_i^{\ast}=\rho_i^{\ast} {M_i}/RT^{\ast}(\simeq\;O(M_i/R)).$

$$ \nabla\cdot\rho_i^{\ast}\overline{{{\mathbf v}}}_i^{\ast}+{\frac{1}{Pe_i}}\,{\frac{M_i}{R}}\left[{\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right]\left[{\frac{1} {T^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}-{\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right]+{\frac{1}{Pe_i}}\rho_i^{\ast}\left[{\frac{1} {T^{\ast}}}\,{\frac{\partial^2T^{\ast}}{\partial x^{*2}}}-{\frac{1} {p^{\ast}_i}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{*2}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)^2\right]=0$$

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1.2 Momentum conservation equation

The momentum conservation equation given in Eq. 15 can be expanded in x and z directions for 2-D analysis. The x-direction momentum equation is given as:

$$ 0=-{\frac{p_iM_i}{RT}}\left[\left(v_{x,i}\,{\frac{\partial v_{x,i}}{\partial x}}+v_{z,i}\,{\frac{\partial v_{x,i}} {\partial z}}\right)+\left({\frac{D_i}{T}}\,{\frac{\partial T} {\partial x}}-{\frac{D_i}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right){\frac{\partial v_{x,i}}{\partial x}}+\left({\frac{D_i} {T}}\,{\frac{\partial T}{\partial z}}-{\frac{D_i}{p_i}}\,{\frac{\partial p_i}{\partial z}}\right){\frac{\partial v_{x,i}}{\partial z}}\right. \left.+\left({\frac{D_i}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{D_i}{p_i}}\,{\frac{\partial p_i} {\partial x}}\right)\left({\frac{D_i}{T}}\,{\frac{\partial^2T} {\partial x^2}}-{\frac{D_i}{p_i}}\,{\frac{\partial^2p_i}{\partial x^2}}-{\frac{D_i}{T^2}}\left({\frac{\partial T}{\partial x}}\right)^2+{\frac{D_i}{p_i^2}}\left({\frac{\partial p_i} {\partial x}}\right)^2\right)\right. \left.+\left({\frac{D_i}{T}}\,{\frac{\partial T}{\partial z}}-{\frac{D_i}{p_i}}\,{\frac{\partial p_i} {\partial z}}\right)\left({\frac{D_i}{T}}\,{\frac{\partial^2T} {\partial z \partial x}}-{\frac{D_i}{p_i}}\,{\frac{\partial^2p_i} {\partial z \partial x}}-{\frac{D_i}{T^2}}\left({\frac{\partial T} {\partial x}}\,{\frac{\partial T}{\partial z}}\right)+{\frac{D_i} {p_i^2}}\left({\frac{\partial p_i}{\partial x}}\,{\frac{\partial p_i} {\partial z}}\right)\right)\right. \left.+v_{x,i}\left({\frac{D_i} {T}}\,{\frac{\partial^2T} {\partial x^2}}-{\frac{D_i} {p_i}}\,{\frac{\partial^2p_i} {\partial x^2}}-{\frac{D_i} {T^2}}\left({\frac{\partial T}{\partial x}}\right)^2+{\frac{D_i} {p_i^2}}\left({\frac{\partial p_i}{\partial x}}\right)^2\right)\right. \left.+v_{z,i}\left({\frac{D_i} {T}}\,{\frac{\partial^2T}{\partial z \partial x}}-{\frac{D_i} {p_i}}\,{\frac{\partial^2p_i}{\partial z \partial x}}-{\frac{D_i} {T^2}}\left({\frac{\partial T}{\partial x}}\,{\frac{\partial T} {\partial z}}\right)+{\frac{D_i}{p_i^2}}\left({\frac{\partial p_i} {\partial x}}\,{\frac{\partial p_i}{\partial z}}\right)\right)\right] -{\frac{\partial p_i}{\partial x}}-{\frac{p} {T}}\sum_{j=1}^n{\frac{x_ix_j} {\fancyscript{D}_{ij}}}\left({\frac{D_i^T}{\rho_i}}-{\frac{D_j^T} {\rho_j}}\right){\frac{\partial T}{\partial x}} +p\sum_{j=1}^n{\frac{x_ix_j} {\fancyscript{D}_{ij}}}\left[(v_{x,j}-v_{x,i})+\left({\frac{D_j} {T}}\,{\frac{\partial T}{\partial x}}-{\frac{D_j}{p_j}}\,{\frac{\partial p_j}{\partial x}}-{\frac{D_i}{T}}\,{\frac{\partial T}{\partial x}}+{\frac{D_i}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right)\right] +\eta_i\left[{\frac{4}{3}}\,{\frac{\partial^2v_{x,i}}{\partial x^2}}+{\frac{\partial^2v_{x,i}}{\partial z^2}}+{\frac{1} {3}}\,{\frac{\partial^2 v_{z,i}}{\partial x\partial z}}\right] +\eta_i\left[{\frac{4}{3}}\left\lbrace {\frac{D_i} {T}}\,{\frac{\partial^3T}{\partial x^3}}-{\frac{D_i}{T^2}}\,{\frac{\partial T}{\partial x}}\,{\frac{\partial^2T}{\partial x^2}}-{\frac{2D_i}{T^2}}\,{\frac{\partial T}{\partial x}}\,{\frac{\partial^2T}{\partial x^2}}+{\frac{2D_i} {T^3}}\left({\frac{\partial T}{\partial x}}\right)^3\right.\right. \left.-{\frac{D_i} {p_i}}\,{\frac{\partial^3p_i}{\partial x^3}}+{\frac{D_i} {p_i^2}}\,{\frac{\partial p_i}{\partial x}}\,{\frac{\partial^2p_i} {\partial x^2}}+{\frac{2D_i}{p_i^2}}\,{\frac{\partial p_i}{\partial x}}\,{\frac{\partial^2p_i}{\partial x^2}}-{\frac{2D_i} {p_i^3}}\left({\frac{\partial p_i}{\partial x}}\right)^3\right\rbrace +\left\lbrace {\frac{D_i} {T}}\,{\frac{\partial^3T}{\partial z^2\partial x}}-{\frac{D_i} {T^2}}\,{\frac{\partial^2T}{\partial z \partial x}}\,{\frac{\partial T} {\partial z}}-{\frac{D_i}{T^2}}\,{\frac{\partial T}{\partial z}}\,{\frac{\partial^2T}{\partial z\partial x}}-{\frac{D_i} {T^2}}\,{\frac{\partial T}{\partial x}}\,{\frac{\partial^2T}{\partial z^2}}+{\frac{2D_i}{T^3}}\,{\frac{\partial T}{\partial x}}\left({\frac{\partial T}{\partial z}}\right)^2\right. -\left. {\frac{D_i} {p_i}}\,{\frac{\partial^3p_i} {\partial z^2\partial x}}+{\frac{D_i} {p_i^2}}\,{\frac{\partial^2p_i}{\partial z \partial x}}\,{\frac{\partial p_i}{\partial z}}+{\frac{D_i}{p_i^2}}\,{\frac{\partial p_i} {\partial z}}\,{\frac{\partial^2p_i}{\partial z\partial x}}+{\frac{D_i}{p_i^2}}\,{\frac{\partial p_i}{\partial x}}\,{\frac{\partial^2p_i}{\partial z^2}}-{\frac{2D_i} {p_i^3}}\,{\frac{\partial p_i}{\partial x}}\left({\frac{\partial p_i} {\partial z}}\right)^2\right\rbrace +{\frac{1}{3}}\left\lbrace{\frac{D_i} {T}}\,{\frac{\partial^3T}{\partial x\partial z^2}}-{\frac{D_i} {T^2}}\,{\frac{\partial T}{\partial x}}\,{\frac{\partial^2T}{\partial z^2}}-{\frac{2D_i}{T^2}}\,{\frac{\partial T}{\partial z}}\,{\frac{\partial^2T}{\partial z\partial x}}+{\frac{2D_i} {T^3}}\,{\frac{\partial T}{\partial x}}\left({\frac{\partial T} {\partial z}}\right)^2\right. -\left.\left.{\frac{D_i} {p_i}}\,{\frac{\partial^3p_i}{\partial x\partial z^2}}+{\frac{D_i} {p_i^2}}\,{\frac{\partial p_i} {\partial x}}\,{\frac{\partial^2p_i} {\partial z^2}}+{\frac{2D_i}{p_i^2}}\,{\frac{\partial p_i}{\partial z}}\,{\frac{\partial^2p_i}{\partial z\partial x}}-{\frac{2D_i} {p_i^3}}\,{\frac{\partial p_i}{\partial x}}\left({\frac{\partial p_i} {\partial z}}\right)^2\right\rbrace\right] $$

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This equation is non-dimensionalized using the scaling parameters defined earlier and the following equation is obtained.

$$-\left[{\frac{U^2}{L}}\underset{\approx O(10^{-6}}{\underbrace{\left(v_{x,i}^{\ast}\,{\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}+v_{z,i}^{\ast}\,{\frac{\partial v_{x,i}^{\ast}} {\partial z^{\ast}}}\right)}}+{\frac{D_iU} {L^2}}\left\lbrace\underset{\approx O(10^{-3})}{\underbrace{\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right){\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}}}+\underset{\approx O(10^{-9})}{\underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right){\frac{\partial v_{x,i}^{\ast}}{\partial z^{\ast}}}}}\right\rbrace+{\frac{D_i^2}{L^3}}\underset{\approx O(1)}{\underbrace{\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)}}\underset{\approx O(1)}{\underbrace{\left\lbrace{\frac{1} {T^{\ast}}}\,{\frac{\partial ^2T^{\ast}} {\partial x^{*2}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial^2p_i^{\ast}} {\partial x^{*2}}}-{\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^2\right\rbrace}}+{\frac{D_i^2}{L^3}}\underset{\approx O(10^{-3})}{\underbrace{\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)}}\underset{\approx O(10^{-3})}{\underbrace{\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}-{\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)\right\rbrace}}+{\frac{D_i}{L^2}}U\underset{\approx O(1)}{v_{z,i}^{\ast}}\underset{\approx O(10^{-3})}{\underbrace{\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}} {\partial x^{\ast}\partial z^{\ast}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)\right\rbrace}}+{\frac{D_i}{L^2}}U\underset{\approx O(10^{-3})}{v_{x,i}^{\ast}}\underset{\approx O(1)}{\underbrace{\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}} {\partial x^{*2}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}} {\partial x^{*2}}}-{\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^2\right\rbrace}}\right]{\frac{\rho U^2M_i}{2T_0R}}\,{\frac{p_i^{\ast}}{T^{\ast}}}+{\frac{\rho U^2}{2}}p^{\ast}\sum_{j=1}^n{\frac{x_ix_j} {\fancyscript{D}_{ij}}}\left[U\underset{\approx O(1)}{\underbrace{\left(v_{x,j}^{\ast}-v_{x,i}^{\ast}\right)}}+\left\lbrace{\frac{D_j} {L}}\underset{\approx O(1)}{\underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}}{\partial x^{\ast}}}\right)}}-{\frac{D_i} {L}}\underset{\approx O(1)}{\underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)}}\right\rbrace\right]-{\frac{\rho U^2}{2L}}\underset{\approx O(1)}{{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}}-{\frac{\rho U^2}{2L}}\underset{\approx O(1)}{{\frac{p^{\ast}}{T^{\ast}}}}\sum_{j=1}^n{\frac{x_ix_j} {\fancyscript{D}_{ij}}}\left({\frac{D_i^T}{\rho_i}}-{\frac{D_j^T} {\rho_j}}\right)\underset{\approx O(1)}{\frac{\partial T^{\ast}} {\partial x^{\ast}}}+\eta_i{{\mathbf S}}_i^{net*}$$

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Where $\eta_i{{\mathbf S}}_i^{net*}$ is given as:

$$ \begin{aligned} \eta_i{\frac{U}{L^2}}\left[{\frac{4}{3}}\underset{\approx O(10^{-3})}{{\frac{\partial^2v_{x,i}^{\ast}}{\partial x^{*2}}}}+\underset{\approx O(10^{-9})}{{\frac{\partial^2v_{x,i}^{\ast}} {\partial z^{*2}}}}+{\frac{1}{3}}\underset{\approx O(10^{-3})}{{\frac{\partial^2v_{z,i}^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}}\right] +\eta_i{\frac{D_i}{L^3}}\left[{\frac{4} {3}}\underbrace{\left\lbrace {\frac{1}{T^{\ast}}}\,{\frac{\partial^3T^{\ast}} {\partial x^{*3}}}-{\frac{1}{T^{*2}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2T^{\ast}}{\partial x^{*2}}}-{\frac{2} {T^{*2}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2T^{\ast}} {\partial x^{*2}}}+{\frac{2}{T^{*3}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^3\right.}_{\approx O(1)} \underbrace{\left.-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial^3p_i^{\ast}}{\partial x^{*3}}}+{\frac{1} {p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{*2}}}+{\frac{2} {p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{*2}}}-{\frac{2} {p_i^{*3}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^3\right\rbrace}_{\approx O(1)} +\underbrace{\left\lbrace {\frac{1} {T^{\ast}}}\,{\frac{\partial^3T^{\ast}}{\partial z^{*2}\partial x^{\ast}}}-{\frac{1} {T^{*2}}}\,{\frac{\partial^2T^{\ast}}{\partial z^{\ast} \partial x^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1} {T^{*2}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial^2T^{\ast}} {\partial z^{\ast}\partial x^{\ast}}}-{\frac{1}{T^{*2}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}\,{\frac{\partial^2T^{\ast}}{\partial z^{*2}}}+{\frac{2} {T^{*3}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\left({\frac{\partial T^{\ast}} {\partial z^{\ast}}}\right)^2\right.}_{\approx O(10^{-6})} -\underbrace{\left.{\frac{1} {p_i^{\ast}}}\,{\frac{\partial^3p_i^{\ast}}{\partial z^{*2}\partial x^{\ast}}}+{\frac{1}{p_i^{*2}}}\,{\frac{\partial^2p_i^{\ast}}{\partial z^{\ast} \partial x^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}+{\frac{1} {p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial z^{\ast}\partial x^{\ast}}}+{\frac{1} {p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial z^{*2}}}-{\frac{2} {p_i^{*3}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\left({\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)^2\right\rbrace}_{\approx O(10^{-6})} +{\frac{1} {3}}\underbrace{\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial^3T^{\ast}} {\partial x^{\ast}\partial z^{*2}}}-{\frac{1}{T^{*2}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2T^{\ast}}{\partial z^{*2}}}-{\frac{2}{T^{*2}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial^2T^{\ast}}{\partial z^{\ast}\partial x^{\ast}}}+{\frac{2} {T^{*3}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}\left({\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)^2\right.}_{\approx O(10^{-6})} -\underbrace{\left.{\frac{1} {p_i^{\ast}}}\,{\frac{\partial^3p_i^{\ast}} {\partial x^{\ast}\partial z^{*2}}}+{\frac{1}{p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial z^{*2}}}+{\frac{2} {p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial z^{\ast}\partial x^{\ast}}}-{\frac{2} {p_i^{*3}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\left({\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)^2\right\rbrace}_{\approx O(10^{-6})}\right]=\eta_i{{\mathbf S}}_i^{net*} \end{aligned} $$

(33)

After neglecting the smaller terms and rearranging the terms, the final x-momentum equation can be given by combining Equations. 32 and 33 as follows.

$$ \begin{aligned} {\frac{M_iU^2}{T_0R}}\,{\frac{p_i^{\ast}} {T^{\ast}}}\left[\left(v_{x,i}^{\ast}\,{\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}+v_{z,i}^{\ast}\,{\frac{\partial v_{x,i}^{\ast}}{\partial z^{\ast}}}\right)\right]+{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}+{\frac{p^{\ast}}{T^{\ast}}}\sum_{j=1}^n{\frac{x_ix_j} {\fancyscript{D}_{ij}}}\left({\frac{D_i^T}{\rho_i}}-{\frac{D_j^T} {\rho_j}}\right){\frac{\partial T^{\ast}}{\partial x^{\ast}}} -p^{\ast}\sum_{j=1}^n{\frac{x_ix_j} {\fancyscript{D}_{ij}}}UL\left(v_{x,j}^{\ast}-v_{x,i}^{\ast}\right)-{\frac{2} {Re_i}}\left[{\frac{4}{3}}\,{\frac{\partial^2v_{x,i}^{\ast}}{\partial x^{*2}}}+{\frac{1}{3}}\,{\frac{\partial^2v_{z,i}^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}\right] +{\frac{M_iU^2}{T_0R}}\,{\frac{p_i^{\ast}}{T^{\ast}}}\left[{\frac{1} {Pe_i}}\,{\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}-{\frac{1} {Pe_i}}\,{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}} +{\frac{1}{Pe_i^2}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}} {\partial x^{*2}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial^2p_i^{\ast}} {\partial x^{*2}}}-{\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^2\right\rbrace +{\frac{1} {Pe_i}}v_{z,i}^{\ast}\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}} {\partial x^{\ast}\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)\right\rbrace+{\frac{1} {Pe_i}}v_{x,i}^{\ast}\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}} {\partial x^{*2}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}} {\partial x^{*2}}}-{\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^2\right\rbrace\right] -p^{\ast}\sum_{j=1}^n{\frac{x_ix_j}{\fancyscript{D}_{ij}}}UL\left[{\frac{1} {Pe_j}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1}{p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}}{\partial x^{\ast}}}\right)-{\frac{1}{Pe_i}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)\right] -{\frac{2}{Re_iPe_i}}\left[{\frac{4}{3}}\left\lbrace {\frac{1} {T^{\ast}}}\,{\frac{\partial^3T^{\ast}} {\partial x^{*3}}}-{\frac{1} {T^{*2}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2T^{\ast}} {\partial x^{*2}}}-{\frac{2}{T^{*2}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2T^{\ast}}{\partial x^{*2}}}+{\frac{2} {T^{*3}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\right)^3-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial^3p_i^{\ast}}{\partial x^{*3}}}+{\frac{1} {p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{*2}}}+{\frac{2} {p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{*2}}}-{\frac{2} {p_i^{*3}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^3\right\rbrace\right]=0 \end{aligned} $$

(34)

The x-momentum equation consists of the conventional and phoretic terms linearly superimposed on each other. For the z-momentum equation, a similar analysis can also be performed. The final z-momentum equation after the analysis is given as:

$$ \begin{aligned} {\frac{M_iU^2}{T_0R}}\,{\frac{p_i^{\ast}} {T^{\ast}}}\left[\left(v_{x,i}^{\ast}\,{\frac{\partial v_{z,i}^{\ast}}{\partial x^{\ast}}}+v_{z,i}^{\ast}\,{\frac{\partial v_{z,i}^{\ast}}{\partial z^{\ast}}}\right)\right]+{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}+{\frac{p^{\ast}}{T^{\ast}}}\sum_{j=1}^n{\frac{x_ix_j} {\fancyscript{D}_{ij}}}\left({\frac{D_i^T}{\rho_i}}-{\frac{D_j^T} {\rho_j}}\right){\frac{\partial T^{\ast}}{\partial z^{\ast}}} -p^{\ast}\sum_{j=1}^n{\frac{x_ix_j} {\fancyscript{D}_{ij}}}UL\left(v_{z,j}^{\ast}-v_{z,i}^{\ast}\right)-{\frac{2} {Re_i}}\,{\frac{\partial ^2v_{z,i}^{\ast}}{\partial x^{*2}}} +{\frac{M_iU^2}{T_0R}}\,{\frac{p_i^{\ast}}{T^{\ast}}}\left[{\frac{1} {Pe_i}}\,{\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial v_{z,i}^{\ast}}{\partial x^{\ast}}}-{\frac{1} {Pe_i}}\,{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial v_{z,i}^{\ast}}{\partial x^{\ast}}}+{\frac{1}{Pe_i^2}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}} {\partial x^{\ast}\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)\right\rbrace+{\frac{1} {Pe_i}}v_{x,i}^{\ast}\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}} {\partial x^{\ast}\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)\right\rbrace+{\frac{1} {Pe_i}}v_{z,i}^{\ast}\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}} {\partial z^{*2}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}} {\partial z^{*2}}}-{\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial z^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)^2\right\rbrace\right]-p^{\ast}\sum_{j=1}^n{\frac{x_ix_j}{\fancyscript{D}_{ij}}}UL\left[{\frac{1} {Pe_j}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1}{p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}}{\partial z^{\ast}}}\right)-{\frac{1}{Pe_i}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)\right] -{\frac{2}{Re_iPe_i}}\left[\left\lbrace {\frac{1} {T^{\ast}}}\,{\frac{\partial^3T^{\ast}}{\partial x^{*2}\partial z^{\ast}}}-{\frac{1} {T^{*2}}}\,{\frac{\partial^2T^{\ast}}{\partial x^{\ast} \partial z^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {T^{*2}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2T^{\ast}} {\partial x^{\ast}\partial z^{\ast}}}-{\frac{1}{T^{*2}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}\,{\frac{\partial^2T^{\ast}}{\partial x^{*2}}}+{\frac{2} {T^{*3}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\right)^2-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial^3p_i^{\ast}}{\partial x^{*2}\partial z^{\ast}}}+{\frac{1}{p_i^{*2}}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{\ast} \partial z^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}+{\frac{1} {p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}+{\frac{1} {p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{*2}}}-{\frac{2} {p_i^{*3}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^2\right\rbrace +{\frac{1}{3}}\left\lbrace{\frac{1} {T^{\ast}}}\,{\frac{\partial^3T^{\ast}}{\partial z^{\ast}\partial x^{*2}}}-{\frac{1} {T^{*2}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial^2T^{\ast}} {\partial x^{*2}}}-{\frac{2}{T^{*2}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2T^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}+{\frac{2} {T^{*3}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\right)^2-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial^3p_i^{\ast}}{\partial z^{\ast}\partial x^{*2}}}+{\frac{1}{p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{*2}}}+{\frac{2} {p_i^{*2}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial^2p_i^{\ast}}{\partial x^{\ast}\partial z^{\ast}}}-{\frac{2} {p_i^{*3}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^2\right\rbrace\right]=0 \end{aligned} $$

(35)

The energy conservation equation

the basic conservation of energy equation given by Eq. 3 can be expanded for the steady-state, and 2-D Stefan tube problem as follows.

$$ 0= \sum_{i=1}^n\underbrace{\eta_i\left[{\frac{4} {3}}\left({\frac{\partial v_{x,i}}{\partial x}}\right)^2-{\frac{4} {3}}\,{\frac{\partial v_{x,i}}{\partial x}}\,{\frac{\partial v_{z,i}} {\partial z}}+2{\frac{\partial v_{x,i}}{\partial z}}\,{\frac{\partial v_{z,i}}{\partial x}}+\left({\frac{\partial v_{z,i}}{\partial x}}\right)^2+\left({\frac{\partial v_{x,i}}{\partial z}}\right)^2+{\frac{4}{3}}\left({\frac{\partial v_{z,i}}{\partial z}}\right)^2\right]}_{{\rm Viscous\;Dissipation}}-\sum_{i=1}^n \underbrace{p_i\left[{\frac{\partial v_{x,i}} {\partial x}}+{\frac{\partial v_{z,i}}{\partial z}}\right]}_{ \rm Pressure\;Work}+\underbrace{{\frac{\partial q_x}{\partial x}}+{\frac{\partial q_z}{\partial z}}}_{{\rm Molecular\;Heat\;Flux}} -\underbrace{\sum_{i=1}^nc_iC_{v,i}\left[v_{x,i}\,{\frac{\partial T}{\partial x}}+v_{z,i}\,{\frac{\partial T}{\partial z}}\right]}_{{\rm Internal\;Energy}}+c_{\rm t}RT\left[\underbrace{\quad\sum_{i=1}^nx_i(v_{x,i}^2+v_{z,i}^2)\sum_j{\frac{x_j} {\fancyscript{D}_{ij}}}-\sum_{i=1}^nx_i\sum_j{\frac{x_j} {\fancyscript{D}_{ij}}}(v_{x,i}v_{x,j}+v_{z,i}v_{z,j})}_{\hbox {Interspecies\;Drag\;Energy}}+\underbrace{\sum_{i=1}^n{\frac{x_i} {T}}\,{\frac{D_i^T}{\rho_i}}\left[v_{x,i}\,{\frac{\partial T}{\partial x}}+v_{z,i}\,{\frac{\partial T}{\partial z}}\right]\sum_j{\frac{x_j} {\fancyscript{D}_{ij}}}+\sum_{i=1}^n{\frac{x_i} {T}}\left[v_{x,i}\,{\frac{\partial T}{\partial x}}+v_{z,i}\,{\frac{\partial T}{\partial z}}\right]\sum_j{\frac{x_j} {\fancyscript{D}_{ij}}}\,{\frac{D_j^T}{\rho_j}}}_{{\rm Interspecies\;Collision\;Energy}}\right]$$

(36)

In the following sections, the individual terms of the energy equations are expanded and the phoretic transport has been incorporated. The terms have been then non-dimensionalized to perform order of magnitude analysis.

2.1 Viscous dissipation

The viscous dissipation term in the energy equation is given as:

$$ W_\eta= \sum_{i=1}^n\eta_i\left[{\frac{4}{3}}\left({\frac{\partial v_{x,i}}{\partial x}}\right)^2-{\frac{4}{3}}\,{\frac{\partial v_{x,i}}{\partial x}}\,{\frac{\partial v_{z,i}}{\partial z}}+2{\frac{\partial v_{x,i}}{\partial z}}\,{\frac{\partial v_{z,i}} {\partial x}}+\left({\frac{\partial v_{z,i}}{\partial x}}\right)^2+\left({\frac{\partial v_{x,i}}{\partial z}}\right)^2+{\frac{4}{3}}\left({\frac{\partial v_{z,i}}{\partial z}}\right)^2\right] $$

(37)

The velocity term in Eq. 37 is replaced by net velocity to incorporate phoretic effects. The modified viscous dissipation terms is as follows:

$$ \begin{aligned} W_\eta=\sum_{i=1}^n-\eta_i\left[{\frac{4} {3}}\left\lbrace\left({\frac{\partial v_{x,i}}{\partial x}}\right)^2+D_i^2\left({\frac{\partial}{\partial x}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1} {p_i}}\,{\frac{\partial p_i}{\partial x}}\right)\right)^2+2D_i{\frac{\partial v_{x,i}}{\partial x}}\,{\frac{\partial}{\partial x}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right)\right\rbrace-{\frac{4} {3}}\left\lbrace{\frac{\partial v_{x,i}}{\partial x}}\,{\frac{\partial v_{z,i}}{\partial z}}+D_i{\frac{\partial v_{x,i}}{\partial x}}\,{\frac{\partial}{\partial z}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial z}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial z}}\right)+D_i{\frac{\partial v_{z,i}}{\partial z}}\,{\frac{\partial} {\partial x}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right)+D_i^2{\frac{\partial} {\partial x}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right){\frac{\partial}{\partial z}}\left({\frac{1} {T}}\,{\frac{\partial T}{\partial z}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial z}}\right)\right\rbrace+2\left\lbrace{\frac{\partial v_{x,i}}{\partial z}}\,{\frac{\partial v_{z,i}}{\partial x}}+D_i{\frac{\partial v_{x,i}}{\partial z}}\,{\frac{\partial} {\partial x}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial z}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial z}}\right)+D_i{\frac{\partial v_{z,i}}{\partial x}}\,{\frac{\partial} {\partial z}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right)+D_i^2{\frac{\partial} {\partial z}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right){\frac{\partial}{\partial x}}\left({\frac{1} {T}}\,{\frac{\partial T}{\partial z}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial z}}\right)\right\rbrace+\left({\frac{\partial v_{z,i}}{\partial x}}\right)^2+D_i^2\left({\frac{\partial}{\partial x}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial z}}-{\frac{1} {p_i}}\,{\frac{\partial p_i}{\partial z}}\right)\right)^2+2D_i{\frac{\partial v_{z,i}}{\partial x}}\,{\frac{\partial}{\partial x}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial z}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial z}}\right) +\left({\frac{\partial v_{x,i}}{\partial z}}\right)^2+D_i^2\left({\frac{\partial}{\partial z}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1} {p_i}}\,{\frac{\partial p_i}{\partial x}}\right)\right)^2+2D_i{\frac{\partial v_{x,i}}{\partial z}}\,{\frac{\partial}{\partial z}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right)+{\frac{4} {3}}\left\lbrace\left({\frac{\partial v_{z,i}}{\partial z}}\right)^2+D_i^2\left({\frac{\partial}{\partial z}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial z}}-{\frac{1} {p_i}}\,{\frac{\partial p_i}{\partial z}}\right)\right)^2+2D_i{\frac{\partial v_{z,i}}{\partial z}}\,{\frac{\partial}{\partial z}}\left({\frac{1}{T}}\,{\frac{\partial T}{\partial z}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial z}}\right)\right\rbrace\right]\\ \end{aligned} $$

(38)

This equation is non-dimensionalized by using the scaling parameters. The order of magnitude analysis for the non-dimensionalized viscous dissipation is as follows:

$$ \begin{aligned} -\sum_{i=1}^n\eta_i{\frac{U^2}{L}}\left[{\frac{4} {3}}\left\lbrace\underset{\approx O(10^{-6})}{\left({\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}\right)^2}+{\frac{1} {Pe_i^2}}\underset{\approx O(1)}{\left({\frac{\partial}{\partial x^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{*}}}\right)\right)^2}+{\frac{2}{Pe_i}}\underset{\approx O(10^{-3})}{{\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial}{\partial x^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)}\right\rbrace-{\frac{4} {3}}\left\lbrace\underset{\approx O(10^{-6})}{{\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial v_{z,i}^{\ast}}{\partial z^{\ast}}}}+{\frac{1}{Pe_i}}\underset{\approx O(10^{-9})}{{\frac{\partial v_{x,i}^{\ast}} {\partial x^{\ast}}}\,{\frac{\partial}{\partial z^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)}+{\frac{1}{Pe_i}}\underset{\approx O(10^{-3})}{{\frac{\partial v_{z,i}^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial}{\partial x^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)}+{\frac{1} {Pe_i^2}}\underset{\approx O(10^{-6})}{{\frac{\partial}{\partial x^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right){\frac{\partial}{\partial z^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)}\right\rbrace+2\left\lbrace\underset{\approx O(10^{-6})}{{\frac{\partial v_{x,i}^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial v_{z,i}^{\ast}}{\partial x^{\ast}}}}+{\frac{1}{Pe_i}}\underset{\approx O(10^{-9})}{{\frac{\partial v_{x,i}^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial}{\partial x^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)}+{\frac{1} {Pe_i}}\underset{\approx O(10^{-3})}{{\frac{\partial v_{z,i}^{\ast}} {\partial x^{\ast}}}\,{\frac{\partial}{\partial z^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)}+{\frac{1} {Pe_i^2}}\underset{\approx O(10^{-6})}{{\frac{\partial}{\partial z^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right){\frac{\partial}{\partial x^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)}\right\rbrace+\underset{\approx O(1)}{\left({\frac{\partial v_{z,i}^{\ast}}{\partial x^{\ast}}}\right)^2}+{\frac{1} {Pe_i^2}}\underset{\approx O(10^{-6})}{\left({\frac{\partial} {\partial x^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{*}}}\right)\right)^2}+{\frac{2}{Pe_i}}\underset{\approx O(10^{-3})}{{\frac{\partial v_{z,i}^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial}{\partial x^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)}+\underset{\approx O(10^{-12})}{\left({\frac{\partial v_{x,i}^{\ast}}{\partial z^{\ast}}}\right)^2}+{\frac{1}{Pe_i^2}}\underset{\approx O(10^{-6})}{\left({\frac{\partial}{\partial z^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{*}}}\right)\right)^2}+{\frac{2}{Pe_i}}\underset{\approx O(10^{-9})}{{\frac{\partial v_{x,i}^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial}{\partial z^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)}+{\frac{4} {3}}\left\lbrace\underset{\approx O(10^{-6})}{\left({\frac{\partial v_{z,i}^{\ast}}{\partial z^{\ast}}}\right)^2}+{\frac{1} {Pe_i^2}}\underset{\approx O(10^{-12})}{\left({\frac{\partial} {\partial z^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{*}}}\right)\right)^2}+{\frac{2}{Pe_i}}\underset{\approx O(10^{-9})}{{\frac{\partial v_{z,i}^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial}{\partial z^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)}\right\rbrace\right] \end{aligned} $$

(39)

After neglecting smaller terms and rearranging the equation, the final viscous heat dissipation term can be expressed as:

$$ W_\eta=-\sum_{i=1}^n\eta_i{\frac{U^2}{L}}\left[{\frac{4} {3}}\,{\frac{1}{Pe_i^2}}\left({\frac{\partial}{\partial x^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)\right)^2+{\frac{8}{3}}\,{\frac{1}{Pe_i}}\,{\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial}{\partial x^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)-{\frac{4}{3}}\,{\frac{1} {Pe_i}}\,{\frac{\partial v_{z,i}^{\ast}}{\partial z^{\ast}}}\,{\frac{\partial} {\partial x^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)+{\frac{2}{Pe_i}}\,{\frac{\partial v_{z,i}^{\ast}} {\partial x^{\ast}}}\,{\frac{\partial}{\partial z^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)+{\frac{2} {Pe_i}}\,{\frac{\partial v_{z,i}^{\ast}}{\partial x^{\ast}}}\,{\frac{\partial} {\partial x^{\ast}}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)+\left({\frac{\partial v_{z,i}^{\ast}}{\partial x^{\ast}}}\right)^2\right]$$

(40)

2.2 Pressure work

The pressure work term in the energy equation (Eq. 36) can be expanded to incorporate phoretic velocity terms as :

$$ W_p=-\sum_{i=1}^np_i\left[{\frac{\partial v_{x,i}}{\partial x}}+{\frac{\partial v_{z,i}}{\partial z}}+{\frac{\partial v_{x,i}^p} {\partial x}}+{\frac{\partial v_{z,i}^p}{\partial z}}\right] $$

(41)

The above equation is expanded by substituting v ^p _i with its expression given in Eq. 9. The expanded equation is further non-dimensionalized to give:

$$W_p=-\sum_{i=1}^n{\frac{\rho_iU^3} {2L}}p_i^{\ast}\left[\underset{\approx O(10^{-3})}{\underbrace{{\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}+{\frac{\partial v_{z,i}^{\ast}}{\partial z^{\ast}}}}} +{\frac{1} {Pe_i}}\underset{\approx O(1)}{\underbrace{\left\lbrace{\frac{1} {T^{\ast}}}\,{\frac{\partial ^2T^{\ast}}{\partial x^{*2}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}}{\partial x^{*2}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}}{\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)^2\right\rbrace}}+{\frac{1} {Pe_i}}\underset{\approx O(10^{-6})}{\underbrace{\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}}{\partial z^{*2}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}} {\partial z^{*2}}}-{\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial z^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)^2\right\rbrace}}\right] $$

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After neglecting the smaller terms the pressure work can be expressed by:

$$ W_p=-\sum_{i=1}^n{\frac{\rho_iU^3}{2L}}p_i^{\ast}\left[{\frac{\partial v_{x,i}^{\ast}}{\partial x^{\ast}}}+{\frac{\partial v_{z,i}^{\ast}}{\partial z^{\ast}}}+{\frac{1}{Pe_i}}\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}}{\partial x^{*2}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}} {\partial x^{*2}}}-{\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^2\right\rbrace\right] $$

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2.3 Molecular heat flux

The molecular heat flux term in the energy equation (Eq. 36) is given as:

$$ Q=\nabla\cdot q^{\prime \prime} $$

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The heat transfer term q″ is modified to account for phoretic transport as per Eq. 11. After expanding the equation, the expression for heat transfer can be obtained as:

$$ \nabla\cdot q^{\prime \prime}=\nabla\cdot \left(-\lambda_{\rm mix}\nabla T+\sum_{i=1}^nc_iC_{pi}Tv_i^p\right) $$

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Which can be further expanded by substituting the phoretic velocity expression from Eq. 9.

$$ \nabla\cdot q^{\prime \prime}=\sum_{i=1}^n{\frac{C_{pi}} {R}}\left[D_i\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right){\frac{\partial p_i}{\partial x}}+D_i\left({\frac{1} {T}}\,{\frac{\partial T}{\partial z}}-{\frac{1}{p_i}}\,{\frac{\partial p_i} {\partial z}}\right){\frac{\partial p_i}{\partial z}}+ p_iD_i\left\lbrace{\frac{1} {T}}\,{\frac{\partial ^2T}{\partial x^2}}-{\frac{1}{p_i}}\,{\frac{\partial ^2p_i}{\partial x^2}}-{\frac{1} {T^2}}\left({\frac{\partial T}{\partial x}}\right)^2+{\frac{1} {p_i^2}}\left({\frac{\partial p_i} {\partial x}}\right)^2\right\rbrace+p_iD_i\left\lbrace{\frac{1} {T}}\,{\frac{\partial ^2T}{\partial z^2}}-{\frac{1}{p_i}}\,{\frac{\partial ^2p_i}{\partial z^2}}-{\frac{1} {T^2}}\left({\frac{\partial T}{\partial z}}\right)^2+{\frac{1} {p_i^2}}\left({\frac{\partial p_i} {\partial z}}\right)^2\right\rbrace\right]-\lambda_{\rm mix}\,{\frac{\partial ^2T}{\partial x^2}}-\lambda_{\rm mix}\,{\frac{\partial ^2T}{\partial z^2}}$$

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The above equation is non-dimensionalized by using the scaling parameters. The order of magnitude analysis of the non-dimensionalized and rearranged equation is presented below:

$$ \nabla\cdot q^{\prime \prime}=\sum_{i=1}^n{\frac{C_{pi}} {R}}\,{\frac{D_i\rho_iU^2}{2L^2}}\left[\underset{\approx O(1)}{\underbrace{\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right){\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}}}+\underset{\approx O(10^{-6})}{\underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right){\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}}}+ \underset{\approx O(1)}{\underbrace{p_i^{\ast}\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}}{\partial x^{*2}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}} {\partial x^{*2}}}-{\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^2\right\rbrace}}+\underset{\approx O(10^{-6})}{\underbrace{p_i^{\ast}\left\lbrace{\frac{1} {T^{\ast}}}\,{\frac{\partial ^2T^{\ast}}{\partial z^{*2}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}}{\partial z^{*2}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)^2\right\rbrace}}\right]-\lambda_{\rm mix}\,{\frac{T_0}{L^2}}\left[\underset{\approx O(1)}{{\frac{\partial ^2T^{\ast}}{\partial x^{*2}}}}+\underset{\approx O(10^{-6})}{{\frac{\partial ^2T^{\ast}}{\partial z^{*2}}}}\right] $$

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After neglecting the smaller terms, the molecular heat transfer term can be finally written as:

$$ \nabla\cdot q^{\prime \prime}=\sum_{i=1}^n{\frac{C_{pi}} {R}}\,{\frac{\rho_iU^3} {2L}}\,{\frac{1}{Pe_i}}\left[\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right){\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}+p_i^{\ast}\left\lbrace{\frac{1}{T^{\ast}}}\,{\frac{\partial ^2T^{\ast}}{\partial x^{*2}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}} {\partial x^{*2}}}-{\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)^2\right\rbrace\right]-\lambda_{\rm mix}\,{\frac{T_0} {L^2}}\,{\frac{\partial ^2T^{\ast}}{\partial x^{*2}}}$$

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2.4 Internal energy

The velocity in internal energy term in Eq. 36 can be replaced by net velocity to account for phoretic transport as:

$$ U=-\sum_{i=1}^nc_iC_{vi}\bar{{{\mathbf v}}}_i\cdot \nabla T =-\sum_{i=1}^nc_iC_{vi}\left[v_{x,i}\,{\frac{\partial T}{\partial x}}+v_{z,i}\,{\frac{\partial T}{\partial z}}+v_{x,i}^p{\frac{\partial T}{\partial x}}+v_{z,i}^p{\frac{\partial T}{\partial z}}\right] $$

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The phoretic velocity is replaced by its expression in Eq. 9 and the equation is non-dimensionalized. The order of magnitude analysis for the non-dimensionalized equation is shown below:

$$ U=-\sum_{i=1}^n{\frac{C_{vi}} {R}}\,{\frac{\rho_iU^3} {2L}}\,{\frac{p_i^{\ast}}{T^{\ast}}}\left[\underset{\approx O(10^{-3})}{\underbrace{\left(v_{x,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}+v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)}}+{\frac{1}{Pe_i}}\underset{\approx O(1)}{\underbrace{\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right){\frac{\partial T^{\ast}} {\partial x^{\ast}}}}}+{\frac{1} {Pe_i}}\underset{\approx O(10^{-6})}{\underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right){\frac{\partial T^{\ast}}{\partial x^{\ast}}}}}\right]$$

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Finally after neglecting the smaller terms, the internal energy can be expressed as:

$$ U=-\sum_{i=1}^n{\frac{C_{vi}}{R}}\,{\frac{\rho_iU^3} {2L}}\,{\frac{p_i^{\ast}}{T^{\ast}}}\left[\left(v_{x,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}+v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}\right)+{\frac{1}{Pe_i}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right){\frac{\partial T^{\ast}}{\partial x^{\ast}}}\right] $$

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2.5 Interspecies drag energy

The interspecies drag term in Eq. 36 can be expanded by replacing the velocity term with net velocity and substituting the phoretic velocity expression. The expanded equation of interspecies drag energy is given as:

$$E_{\rm drag}=p\left[\sum_{i=1}^nx_i\left\lbrace v_{x,i}^2+{\frac{D_i^2}{T^2}}\left({\frac{\partial T}{\partial x}}\right)^2+{\frac{D_i^2}{p_i^2}}\left({\frac{\partial p_i} {\partial x}}\right)^2-{\frac{2D_i^2}{Tp_i}}\,{\frac{\partial T} {\partial x}}\,{\frac{\partial p_i}{\partial x}}+v_{z,i}^2+{\frac{D_i^2} {T^2}}\left({\frac{\partial T}{\partial z}}\right)^2+{\frac{D_i^2} {p_i^2}}\left({\frac{\partial p_i}{\partial z}}\right)^2-{\frac{2D_i^2}{Tp_i}}\,{\frac{\partial T}{\partial z}}\,{\frac{\partial p_i}{\partial z}}+2D_iv_{x,i}\left({\frac{1} {T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right)+2D_iv_{z,i}\left({\frac{1} {T}}\,{\frac{\partial T} {\partial z}}-{\frac{1}{p_i}}\,{\frac{\partial p_i} {\partial z}}\right)\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}-\sum_{i=1}^nx_i\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}\left\lbrace v_{x,i}D_j\left({\frac{1} {T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_j}}\,{\frac{\partial p_j}{\partial x}}\right)+v_{x,j}D_i\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right)+v_{x,i}v_{x,j}+D_iD_j\left({\frac{1} {T}}\,{\frac{\partial T} {\partial x}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial x}}\right)\left({\frac{1}{T}}\,{\frac{\partial T}{\partial x}}-{\frac{1}{p_j}}\,{\frac{\partial p_j}{\partial x}}\right)+v_{z,i}D_j\left({\frac{1} {T}}\,{\frac{\partial T}{\partial z}}-{\frac{1}{p_j}}\,{\frac{\partial p_j}{\partial z}}\right)+v_{z,j}D_i\left({\frac{1}{T}}\,{\frac{\partial T}{\partial z}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial z}}\right)+v_{z,i}v_{z,j}+D_iD_j\left({\frac{1}{T}}\,{\frac{\partial T}{\partial z}}-{\frac{1} {p_i}}\,{\frac{\partial p_i}{\partial z}}\right)\left({\frac{1}{T}}\,{\frac{\partial T}{\partial z}}-{\frac{1}{p_j}}\,{\frac{\partial p_j}{\partial z}}\right)\right\rbrace\right]$$

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The above equation is non-dimensionalized by proper scaling parameters. The orders of magnitude of different terms of the equation are given as:

$$ \begin{aligned} E_{\rm drag}={\frac{\rho U^2} {2}}p^{\ast}\left[\sum_{i=1}^nx_iU_i^2\left\lbrace \underset{\approx O(10^{-6})}{ \underbrace{v_{x,i}^{*2}}}+{\frac{1} {Pe_i^2}}\underset{\approx O(1)}{ \underbrace{\left({\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1}{p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)^2-{\frac{2} {T^{\ast}p_i^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)}}+\underset{\approx O(1)}{ \underbrace{v_{z,i}^{*2}}}+{\frac{1}{Pe_i^2}} \underset{\approx O(10^{-6})}{ \underbrace{\left({\frac{1}{T^{*2}}}\left({\frac{\partial T^{\ast}}{\partial z^{\ast}}}\right)^2+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)^2-{\frac{2} {T^{\ast}p_i^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)}}+{\frac{2} {Pe_i}}\underset{\approx O(10^{-3})}{ \underbrace{\left(v_{x,i}^{\ast}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)+v_{z,i}^{\ast}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)\right)}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}-\sum_{i=1}^nx_i\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}U_iU_j\left\lbrace {\frac{v_{x,i}^{\ast}} {Pe_j}}\underset{\approx O(10^{-3})}{ \underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}}{\partial x^{\ast}}}\right)}}+{\frac{v_{x,j}}{Pe_i}}\underset{\approx O(10^{-3})}{ \underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)}}+\underset{\approx O(10^{-6})}{ \underbrace{v_{x,i}^{\ast}v_{x,j}^{\ast}}}+{\frac{1} {Pe_iPe_j}}\underset{\approx O(1)}{ \underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial x^{\ast}}}\right)\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}}{\partial x^{\ast}}}\right)}}+{\frac{v_{z,i}^{\ast}} {Pe_j}}\underset{\approx O(10^{-3})}{ \underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1} {p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}} {\partial z^{\ast}}}\right)}}+{\frac{v_{z,j}^{\ast}}{Pe_i}}\underset{\approx O(10^{-3})}{ \underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)}}+\underset{\approx O(1)}{ \underbrace{v_{z,i}^{\ast}v_{z,j}^{\ast}}}+{\frac{1} {Pe_iPe_j}}\underset{\approx O(10^{-6})}{ \underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1}{p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}}{\partial z^{\ast}}}\right)}}\right\rbrace\right] \end{aligned} $$

(53)

After neglecting the smaller terms, the interspecies drag energy can be finally expressed as:

$$ \begin{aligned} E_{\rm drag}={\frac{\rho U^2} {2}}p^{\ast}\left[\sum_{i=1}^nx_iU_i^2\left\lbrace {\frac{1} {Pe_i^2}}\left({\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)^2-{\frac{2} {T^{\ast}p_i^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)+v_{z,i}^{*2}+{\frac{2} {Pe_i}}\left(v_{x,i}^{\ast}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1}{p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)+v_{z,i}^{\ast}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}}{\partial z^{\ast}}}\right)\right)\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}} -\sum_{i=1}^nx_i\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}U_iU_j\left\lbrace {\frac{v_{x,i}^{\ast}} {Pe_j}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1}{p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}}{\partial x^{\ast}}}\right)+{\frac{v_{x,j}}{Pe_i}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)+{\frac{1} {Pe_iPe_j}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}} {\partial x^{\ast}}}\right)+v_{z,i}^{\ast}v_{z,j}^{\ast}+{\frac{v_{z,i}^{\ast}} {Pe_j}}\left({\frac{1}{T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1}{p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}}{\partial z^{\ast}}}\right)+{\frac{v_{z,j}^{\ast}}{Pe_i}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}}{\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)\right\rbrace\right] \end{aligned} $$

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2.6 Interspecies collision energy

The interspecies collision terms in Eq. 36 can be expanded to account for phoretic transport as:

$$ E_{\rm col}=p\left[\sum_{i=1}^nx_i{\frac{D_i^T} {\rho_iT}}\left\lbrace v_{x,i}\,{\frac{\partial T}{\partial x}}+D_i\left({\frac{1} {T}}\,{\frac{\partial T}{\partial x}}-{\frac{1} {p_i}}\,{\frac{\partial p_i} {\partial x}}\right){\frac{\partial T} {\partial x}}+v_{z,i}\,{\frac{\partial T} {\partial z}}+D_i\left({\frac{1}{T}}\,{\frac{\partial T} {\partial z}}-{\frac{1}{p_i}}\,{\frac{\partial p_i}{\partial z}}\right){\frac{\partial T}{\partial z}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}-\sum_{i=1}^nx_i{\frac{1}{T}}\left\lbrace v_{x,i}\,{\frac{\partial T}{\partial x}}+D_i\left({\frac{1} {T}}\,{\frac{\partial T}{\partial x}}-{\frac{1} {p_i}}\,{\frac{\partial p_i} {\partial x}}\right){\frac{\partial T}{\partial x}}+v_{z,i}\,{\frac{\partial T}{\partial z}}+D_i\left({\frac{1} {T}}\,{\frac{\partial T} {\partial z}}-{\frac{1} {p_i}}\,{\frac{\partial p_i} {\partial z}}\right){\frac{\partial T} {\partial z}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}\,{\frac{D_j^T} {\rho_j}}\right]$$

(55)

This equation is non-dimensionalized. The order of magnitude analysis of the equation is given as:

$$E_{\rm col}={\frac{\rho U^3} {2L}}p^{\ast}\left[\sum_{i=1}^nx_i{\frac{D_i^T} {\rho_iT^{\ast}}}\left\lbrace \underset{\approx O(10^{-3})}{\underbrace{v_{x,i^{\ast}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}}}+{\frac{1} {Pe_i}} \underset{\approx O(1)}{\underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right){\frac{\partial T^{\ast}} {\partial x^{\ast}}}}}+ \underset{\approx O(10^{-3})}{\underbrace{v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}}}+{\frac{1} {Pe_i}} \underset{\approx O(10^{-6})}{\underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right){\frac{\partial T^{\ast}} {\partial z^{\ast}}}}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}-\sum_{i=1}^nx_i{\frac{1} {T^{\ast}}}\left\lbrace \underset{\approx O(10^{-3})}{\underbrace{v_{x,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}}}+{\frac{1} {Pe_i}} \underset{\approx O(1)}{\underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right){\frac{\partial T^{\ast}} {\partial x^{\ast}}}}}+ \underset{\approx O(10^{-3})}{\underbrace{v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}}}+{\frac{1} {Pe_i}} \underset{\approx O(10^{-6})}{\underbrace{\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right){\frac{\partial T^{\ast}} {\partial z^{\ast}}}}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}\,{\frac{D_j^T} {\rho_j}}\right]$$

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After neglecting the smaller terms, the final interspecies collision energy can be expressed as:

$$ E_{\rm col}={\frac{\rho U^3} {2L}}p^{\ast}\left[\sum_{i=1}^nx_i{\frac{D_i^T} {\rho_iT^{\ast}}}\left\lbrace v_{x,i^{\ast}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}+{\frac{1} {Pe_i}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right){\frac{\partial T^{\ast}} {\partial x^{\ast}}}+ v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}-\sum_{i=1}^nx_i{\frac{1} {T^{\ast}}}\left\lbrace v_{x,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}+{\frac{1} {Pe_i}} \left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right){\frac{\partial T^{\ast}} {\partial x^{\ast}}}+ v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}\,{\frac{D_j^T} {\rho_j}}\right] $$

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2.7 Final energy equation

All the significant terms of Eq. 36 can be combined to form the complete energy equation.

$$ \begin{aligned} 0=-\sum_{i=1}^n{\frac{\rho_iU^3} {2L}}p_i^{\ast}\left[{\frac{\partial v_{x,i}^{\ast}} {\partial x^{\ast}}}+{\frac{\partial v_{z,i}^{\ast}} {\partial z^{\ast}}}\right]-\sum_{i=1}^n\eta_i{\frac{U^2} {L}}\left({\frac{\partial v_{z,i}^{\ast}} {\partial x^{\ast}}}\right)^2-\lambda_{\rm mix}\,{\frac{T_0} {L^2}}\,{\frac{\partial ^2T^{\ast}} {\partial x^{*2}}}-\sum_{i=1}^n{\frac{C_{vi}} {R}}\,{\frac{\rho_iU^3} {2L}}\,{\frac{p_i^{\ast}} {T^{\ast}}}\left[\left(v_{x,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}+v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}\right)\right]+{\frac{\rho U^4} {2}}p^{\ast}\left[\sum_{i=1}^nx_iv_{z,i}^{*2}\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}-\sum_{i=1}^nx_i\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}v_{z,i}^{\ast}v_{z,j}^{\ast}\right]+{\frac{\rho U^3} {2L}}\,{\frac{p^{\ast}} {T^{\ast}}}\left[\sum_{i=1}^nx_i{\frac{D_i^T} {\rho_i}}\left\lbrace v_{x,i^{\ast}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}+ v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}-\sum_{i=1}^nx_i\left\lbrace v_{x,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}+ v_{z,i}^{\ast}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}\,{\frac{D_j^T} {\rho_j}}\right]-\sum_{i=1}^n\eta_i{\frac{U^2} {L}}\left[{\frac{4} {3}}\,{\frac{1} {Pe_i^2}}\left({\frac{\partial} {\partial x^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)\right)^2+{\frac{8} {3}}\,{\frac{1} {Pe_i}}\,{\frac{\partial v_{x,i}^{\ast}} {\partial x^{\ast}}}\,{\frac{\partial} {\partial x^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)-{\frac{4} {3}}\,{\frac{1} {Pe_i}}\,{\frac{\partial v_{z,i}^{\ast}} {\partial z^{\ast}}}\,{\frac{\partial} {\partial x^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)+{\frac{2} {Pe_i}}\,{\frac{\partial v_{z,i}^{\ast}} {\partial x^{\ast}}}\,{\frac{\partial} {\partial z^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)+{\frac{2} {Pe_i}}\,{\frac{\partial v_{z,i}^{\ast}} {\partial x^{\ast}}}\,{\frac{\partial} {\partial x^{\ast}}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)\right]-\sum_{i=1}^n{\frac{\rho_iU^3} {2L}}p_i^{\ast}\left[{\frac{1} {Pe_i}}\left\lbrace{\frac{1} {T^{\ast}}}\,{\frac{\partial ^2T^{\ast}} {\partial x^{*2}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}} {\partial x^{*2}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)^2\right\rbrace\right]+\sum_{i=1}^n{\frac{M_iC_{pi}} {R}}\,{\frac{\rho_iU^3} {2L}}\,{\frac{1} {Pe_i}}\left[\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right){\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}+p_i^{\ast}\left\lbrace{\frac{1} {T^{\ast}}}\,{\frac{\partial ^2T^{\ast}} {\partial x^{*2}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial ^2p_i^{\ast}} {\partial x^{*2}}}-{\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)^2\right\rbrace\right]-\sum_{i=1}^n{\frac{C_{vi}} {R}}\,{\frac{\rho_iU^3} {2L}}\,{\frac{p_i^{\ast}} {T^{\ast}}}\left[{\frac{1} {Pe_i}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right){\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right]+{\frac{\rho U^4} {2}}p^{\ast}\left[\sum_{i=1}^nx_i\left\lbrace {\frac{1} {Pe_i^2}}\left({\frac{1} {T^{*2}}}\left({\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right)^2+{\frac{1} {p_i^{*2}}}\left({\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)^2-{\frac{2} {T^{\ast}p_i^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)+{\frac{2} {Pe_i}}\left(v_{x,i}^{\ast}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)+v_{z,i}^{\ast}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)\right)\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}} -\sum_{i=1}^nx_i\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}\left\lbrace {\frac{v_{x,i}^{\ast}} {Pe_j}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}} {\partial x^{\ast}}}\right)+{\frac{v_{x,j}} {Pe_i}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)+{\frac{1} {Pe_iPe_j}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right)\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}} {\partial x^{\ast}}}\right)+{\frac{v_{z,i}^{\ast}} {Pe_j}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1} {p_j^{\ast}}}\,{\frac{\partial p_j^{\ast}} {\partial z^{\ast}}}\right)+{\frac{v_{z,j}^{\ast}} {Pe_i}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial z^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial z^{\ast}}}\right)\right\rbrace\right]+{\frac{\rho U^3} {2L}}\,{\frac{p^{\ast}} {T^{\ast}}}\left[\sum_{i=1}^nx_i{\frac{D_i^T} {\rho_i}}\left\lbrace {\frac{1} {Pe_i}}\left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right){\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}-\sum_{i=1}^nx_i\left\lbrace {\frac{1} {Pe_i}} \left({\frac{1} {T^{\ast}}}\,{\frac{\partial T^{\ast}} {\partial x^{\ast}}}-{\frac{1} {p_i^{\ast}}}\,{\frac{\partial p_i^{\ast}} {\partial x^{\ast}}}\right){\frac{\partial T^{\ast}} {\partial x^{\ast}}}\right\rbrace\sum_{j=1}^n{\frac{x_j} {\fancyscript{D}_{ij}}}\,{\frac{D_j^T} {\rho_j}}\right] \end{aligned} $$

(58)