Loading docs/source/FluidEquations.rst +5 −5 Original line number Diff line number Diff line Loading @@ -47,18 +47,18 @@ The conservation of fluid momentum is: = \nabla \cdot \tau + M_{sg} + \varepsilon_g \rho_g g where :math:`M_{sg} = - M_{gs}` is the generalized interfacial momentum transfer from the solid particles to the gas-phase to the gas-phase. Like :math:`\varepsilon_s`, the solid particles to the fluid-phase. Like :math:`\varepsilon_s`, :math:`M_{gs}` is determined from the L-E transfer kernel by setting :math:`A_i = F_{gi}`, where :math:`F_{gi}` is the force due to the gas-phase on the ith particle. Following where :math:`F_{gi}` is the force due to the fluid-phase on the ith particle. Following MFiX classic (and many other CFD-DEM codes designed for high density ratio gas-solids flows), only buoyancy (pressure gradient) and steady drag are considered: .. math:: F_{gi} = - \mathcal{V}_i \nabla p_g - \frac{1}{2} C_D \rho_g \boldsymbol{V}_{ig} - \left|\boldsymbol{V}_{ig}\right| A_i^{(proj)} - \frac{1}{2} C_D \rho_g \boldsymbol{V}_{ig} \left|\boldsymbol{V}_{ig}\right| A_i^{(proj)} where :math:`\boldsymbol{V}_{ig} = \boldsymbol{V}_i - \boldsymbol{U}_g ( \boldsymbol{X}_i )` is the velocity of ith particle relative to the gas-phase (at the particle position is the velocity of ith particle relative to the fluid-phase (at the particle position :math:`\boldsymbol{X}_i`). :math:`F_{gi}` is closed by the specification of a drag coefficient, :math:`C_D`. Currently, MFiX-Exa includes Wen-Yu, Gidaspow and BVK2 drag laws. Loading @@ -73,7 +73,7 @@ coefficient, :math:`C_D`. Currently, MFiX-Exa includes Wen-Yu, Gidaspow and BVK2 In chemical engineering literature, it is common to lump all drag-related terms of :math:`M_{gs}` into :math:`\beta`. With this simplification and some re-arrangement, the flud momentum takes the more convenient form: the fluid momentum takes the more convenient form: .. math:: \frac{ \partial (\varepsilon_g \rho_g U_g)}{\partial t} + \nabla \cdot (\varepsilon_g \rho_g U_g U_g) + \varepsilon_g \nabla p_g Loading Loading
docs/source/FluidEquations.rst +5 −5 Original line number Diff line number Diff line Loading @@ -47,18 +47,18 @@ The conservation of fluid momentum is: = \nabla \cdot \tau + M_{sg} + \varepsilon_g \rho_g g where :math:`M_{sg} = - M_{gs}` is the generalized interfacial momentum transfer from the solid particles to the gas-phase to the gas-phase. Like :math:`\varepsilon_s`, the solid particles to the fluid-phase. Like :math:`\varepsilon_s`, :math:`M_{gs}` is determined from the L-E transfer kernel by setting :math:`A_i = F_{gi}`, where :math:`F_{gi}` is the force due to the gas-phase on the ith particle. Following where :math:`F_{gi}` is the force due to the fluid-phase on the ith particle. Following MFiX classic (and many other CFD-DEM codes designed for high density ratio gas-solids flows), only buoyancy (pressure gradient) and steady drag are considered: .. math:: F_{gi} = - \mathcal{V}_i \nabla p_g - \frac{1}{2} C_D \rho_g \boldsymbol{V}_{ig} - \left|\boldsymbol{V}_{ig}\right| A_i^{(proj)} - \frac{1}{2} C_D \rho_g \boldsymbol{V}_{ig} \left|\boldsymbol{V}_{ig}\right| A_i^{(proj)} where :math:`\boldsymbol{V}_{ig} = \boldsymbol{V}_i - \boldsymbol{U}_g ( \boldsymbol{X}_i )` is the velocity of ith particle relative to the gas-phase (at the particle position is the velocity of ith particle relative to the fluid-phase (at the particle position :math:`\boldsymbol{X}_i`). :math:`F_{gi}` is closed by the specification of a drag coefficient, :math:`C_D`. Currently, MFiX-Exa includes Wen-Yu, Gidaspow and BVK2 drag laws. Loading @@ -73,7 +73,7 @@ coefficient, :math:`C_D`. Currently, MFiX-Exa includes Wen-Yu, Gidaspow and BVK2 In chemical engineering literature, it is common to lump all drag-related terms of :math:`M_{gs}` into :math:`\beta`. With this simplification and some re-arrangement, the flud momentum takes the more convenient form: the fluid momentum takes the more convenient form: .. math:: \frac{ \partial (\varepsilon_g \rho_g U_g)}{\partial t} + \nabla \cdot (\varepsilon_g \rho_g U_g U_g) + \varepsilon_g \nabla p_g Loading