Table 1 Governing equations of the models47,49.
Governing equations | Single-phase model | Two-phase model |
|---|---|---|
Continuity | \(\nabla \cdot u=0\) | \(\frac{\partial {\rho }_{m}}{\partial t}+\nabla \cdot \left({\rho }_{m}{u}_{m}\right)=0\) |
Momentum | \(\frac{\partial u}{\partial t}+\nabla \cdot \left(uu\right)=-\nabla \left(\frac{P}{{\rho }_{m}}\right)+\nabla \cdot \left(\frac{{\mu }_{m}}{{\rho }_{m}}\nabla u\right)+{g}_{k}+{S}_{m}\) | \(\frac{\partial \left({\rho }_{m}{u}_{m}\right)}{\partial t}+\nabla \cdot \left({\rho }_{m}{u}_{m}{u}_{m}\right)=-\nabla P+\nabla \cdot \left({\mu }_{m}\nabla {u}_{m}\right)+\nabla \cdot \left(\begin{array}{c}\varphi {\rho }_{p}{u}_{pm}{u}_{pm}+\\ \varphi {\rho }_{p}{u}_{T}{u}_{T}+\\ \varphi {\rho }_{p}{u}_{B}{u}_{B}\\ \left(1-\varphi \right){\rho }_{bf}{u}_{bfm}{u}_{bfm}\end{array}\right)+{\rho }_{m}{g}_{k}+{S}_{m}\) \({u}_{pm}=\frac{{\rho }_{bf}}{{\rho }_{m}}{u}_{s}{10}^{-A\varphi }\), \({u}_{bfm}=\frac{\varphi {\rho }_{p}}{\left(1-\varphi \right){\rho }_{f}}{u}_{pm}\), \({u}_{s}=\frac{{d}_{p}^{2}\left({\rho }_{p}-{\rho }_{bf}\right)}{18{\mu }_{bf}}g\), \({u}_{B}=-{D}_{B}\nabla \varphi\),\({u}_{T}=-{S}_{T}\frac{{\mu }_{m}}{{\rho }_{m}}\frac{\nabla {T}_{m}}{{T}_{m}}\) |
Energy | \(\frac{\partial T}{\partial t}+\nabla \cdot \left(Tu\right)=\nabla \cdot \left(\frac{{k}_{m}}{{\rho }_{m}{C}_{pm}}\nabla T\right)+{S}_{h}-\nabla \cdot {q}_{t}\) | \(\frac{\partial \left({\rho }_{m}{C}_{pm}{T}_{m}\right)}{\partial t}+\nabla \cdot \left({\rho }_{m}{u}_{m}{C}_{pm}{T}_{m}\right)=\nabla \cdot \left({k}_{m}\nabla T\right)-{C}_{pp}{J}_{p}\cdot \nabla {T}_{m}+{S}_{h}-\nabla \cdot {q}_{t}\) \({J}_{p}={\rho }_{p}\left({D}_{B}\nabla \varphi +{D}_{T}\frac{\nabla {T}_{m}}{{T}_{m}}\right)={\rho }_{p}\left({u}_{B}+\varphi {u}_{T}\right)\), \({D}_{B}=\frac{{k}_{B}{T}_{m}}{3\pi {\mu }_{bf}{d}_{p}}\), \({D}_{T}=\beta \frac{{\mu }_{m}}{{\rho }_{m}}\varphi\) |
Nanoparticle transport | Â | \(\frac{\partial \left({\rho }_{p}\varphi \right)}{\partial t}+\nabla \cdot \left(\varphi {\rho }_{p}{u}_{m}\right)=-\nabla \cdot \left(\varphi {\rho }_{p}{u}_{pm}+\varphi {\rho }_{p}{u}_{T}+\varphi {\rho }_{p}{u}_{B}\right)+{S}_{pm}\) with \({S}_{pm}=-\left({u}_{m}+{u}_{pm}+{u}_{T}+{u}_{B}\right)C\frac{{\left(1-\alpha \right)}^{2}}{{\alpha }^{3}+b}\) |
Liquid fraction | \(\alpha =0.5\cdot \mathrm{erf}\left(\frac{4\left(T-{T}_{me}\right)}{{T}_{l}-{T}_{s}}\right)+0.5\) | \(\alpha =0.5\cdot \mathrm{erf}\left(\frac{4\left({T}_{m}-{T}_{me}\right)}{{T}_{l}-{T}_{s}}\right)+0.5\) |
Closure Parameters | ||
Buoyancy term | \({g}_{k}=\left[1-{\beta }_{m}\left(T-{T}_{ref}\right)\right]g\) | \({g}_{k}=\left[1-{\beta }_{m}\left({T}_{m}-{T}_{ref}\right)\right]g.\) |
Enthalpy source term | \({S}_{h}=-\rho L\frac{4.\mathrm{exp}\left({\left(\frac{4\left(T-{T}_{me}\right)}{{T}_{l}-{T}_{s}}\right)}^{2}\right)}{\left({T}_{l}-{T}_{s}\right)\sqrt{\pi }}\cdot \left(\frac{\partial T}{\partial t}+u\nabla T\right)\) | \({S}_{h}=-\rho L\frac{4.\mathrm{exp}\left({\left(\frac{4\left({T}_{m}-{T}_{me}\right)}{{T}_{l}-{T}_{s}}\right)}^{2}\right)}{\left({T}_{l}-{T}_{s}\right)\sqrt{\pi }}\cdot \left(\frac{\partial {T}_{m}}{\partial t}+{u}_{m}\nabla {T}_{m}\right)\), |
Momentum source term | \({S}_{m}=-uC\frac{{\left(1-\alpha \right)}^{2}}{{\alpha }^{3}+b}\) | \({S}_{m}=-{u}_{m}C\frac{{\left(1-\alpha \right)}^{2}}{{\alpha }^{3}+b}\) |
