.. _sec.units.cyclone: Cyclone ======= Solids-gas separation according to Muschelknautz .. image:: ../static/images/003_models/cyclone.png :width: 200px :alt: cyclone :align: center | Constant geometric parameters .. math:: r_{o} = 0.5d_{o} .. math:: r_{f} = 0.5d_{f} .. math:: r_{exit} = 0.5d_{exit} .. math:: b_{e} = \begin{cases} \text{user-defined} & \text{rect slot, full/half spiral entry} \\ r_{o} - r_{core} & \text{axial entry} \end{cases} .. math:: r_{e} = \begin{cases} r_{o} - 0.5b_{e} & \text{rect slot, axial entry} \\ r_{o} + 0.5b_{e} & \text{full spiral entry} \\ r_{o} & \text{half spiral entry} \\ \end{cases} .. math:: {\overline{r}}_{con} = 0.5\left( r_{o} + r_{exit} \right) .. math:: r_{exit,eff} = \begin{cases} r_{f} & r_{exit} \leq r_{f} \\ r_{exit} & r_{exit} > r_{f} \\ \end{cases} .. math:: \beta = \frac{b_{e}}{r_{o}} .. math:: h_{con} = h_{tot} - h_{cyl} .. math:: h_{con,eff} = \left( \frac{r_{o} - r_{exit,eff}}{r_{o} - r_{exit}} \right)h_{con} .. math:: h_{sep} = h_{cyl} + h_{con,eff} - h_{f} .. math:: a = \begin{cases} \text{-} & \text{rect slot, full/half spiral entry} \\ \sin(\delta)\frac{\pi\left( r_{o} + r_{core} \right)}{N_{b}} - d_{b} & \text{axial entry} \\ \end{cases} .. math:: A_{cyl} = 2\pi r_{o}h_{cyl} .. math:: A_{con} = \pi\left( r_{o} + r_{exit,eff} \right)\sqrt{h_{con,eff}^{2} + \left( r_{o} - r_{exit,eff} \right)^{2}} .. math:: A_{top} = \pi r_{o}^{2} - \pi r_{f}^{2} .. math:: A_{f} = 2\pi r_{f}h_{f} .. math:: A_{tot} = \begin{cases} A_{cyl} + A_{con} + A_{f} + A_{top} & \text{rect slot, axial entry} \\ A_{cyl} + A_{con} + A_{f} + A_{top} - \varepsilon r_{o}h_{e} & \text{full/half spiral entry} \\ \end{cases} .. math:: A_{con/2} = \pi\left( r_{o} + {\overline{r}}_{con} \right)\sqrt{\left( \frac{h_{con}}{2} \right)^{2} + \left( r_{o} - {\overline{r}}_{con} \right)^{2}} .. math:: A_{sed} = A_{cyl} + A_{con/2} .. math:: A_{e1} = \frac{2\pi r_{o}h_{e}}{2} .. math:: A_{sp} = \begin{cases} \text{-} & \text{rect slot, axial entry} \\ \varepsilon\left( \frac{b + 2r_{o}}{2}\left( b_{e} + h_{e} \right) \right) & \text{full spiral entry} \\ \varepsilon r_{o}\left( b_{e} + h_{e} \right) & \text{half spiral entry} \\ \end{cases} Operational parameters .. math:: {\dot{V}}_{in,g} = \frac{{\dot{m}}_{in,g}}{\rho_{g}} .. math:: \mu_{in} = \frac{{\dot{m}}_{in,s}}{{\dot{m}}_{in,g}} .. math:: \lambda_{s} = \begin{cases} \lambda_{0}\left( 1 + 2\sqrt{\mu_{in}} \right) & \mu_{in} \leq 1 \\ \lambda_{0}\left( 1 + 3\sqrt{\mu_{in}} \right) & \mu_{in} > 1 \\ \end{cases} .. math:: \alpha = \begin{cases} \frac{1}{\beta}\left( 1 - \sqrt{1 + 4\left\lbrack \left( \frac{\beta}{2} \right)^{2} - \left( \frac{\beta}{2} \right) \right\rbrack\sqrt{1 - \frac{1 - \beta^{2}}{1 + \mu_{in}}\left( 2\beta - \beta^{2} \right)}} \right) & \text{rect slot, full/half spiral entry} \\ \begin{cases} 0.85 & \text{simple straight blades} \\ 0.95 & \text{curved blades} \\ 1.05 & \text{curved and twisted blades} \\ \end{cases} & \text{axial entry} \\ \end{cases} Geometric parameters .. math:: {\overline{r}}_{e} = r_{o} - \frac{\alpha b_{e}}{2} .. math:: {\overline{r}}_{z} = \sqrt{{\overline{r}}_{e}{\overline{r}}_{con}} Velocities .. math:: v_{e} = \begin{cases} {\dot{V}}_{in,g}/\left( b_{e}h_{e} \right) & \text{rect slot, full/half spiral entry} \\ {\dot{V}}_{in,g}/\left( ab_{e}N_{b} \right) & \text{axial entry} \\ \end{cases} .. math:: w_{50} = \frac{0.5\left( 0.9{\dot{V}}_{in,g} \right)}{A_{sed}} .. math:: u_{o} = \begin{cases} \frac{v_{e}\frac{r_{e}}{r_{o}}}{\alpha} & \text{rect slot entry} \\ \frac{v_{e}\frac{r_{e}}{r_{o}}}{1 + \frac{\lambda_{s}}{2}\frac{A_{sp}}{{\dot{V}}_{in,g}}v_{e}\sqrt{\frac{r_{e}}{r_{o}}}\ } & \text{full/half spiral entry} \\ \frac{v_{e}\cos(\delta)\frac{r_{e}}{r_{o}}}{\alpha} & \text{axial entry} \\ \end{cases} .. math:: u_{f} = \frac{u_{o}\frac{r_{o}}{r_{f}}}{1 + \frac{\lambda_{s}}{2}\frac{A_{tot}}{{\dot{V}}_{in,g}}u_{o}\sqrt{\frac{r_{o}}{r_{f}}}} .. math:: u_{e} = \frac{u_{o}\frac{r_{o}}{{\overline{r}}_{e}}\ }{1 + \frac{\lambda_{s}}{2}\frac{A_{e1}}{0.9{\dot{V}}_{in,g}}u_{o}\sqrt{\frac{r_{o}}{{\overline{r}}_{e}}}} .. math:: u_{con} = \frac{u_{o}\frac{r_{o}}{{\overline{r}}_{con}}\ }{1 + \frac{\lambda_{s}}{2}\frac{A_{sed}}{0.9{\dot{V}}_{in,g}}u_{o}\sqrt{\frac{r_{o}}{{\overline{r}}_{con}}}} Mass separation between main and secondary streams .. math:: n = \frac{\ln\frac{u_{f}}{u_{o}}}{\ln\frac{r_{o}}{r_{f}\ }} .. math:: {\dot{V}}_{\sec} = {\dot{V}}_{in,g}\left( 0.0497 + 0.0684n + 0.0949n^{2} \right) .. math:: w_{split} = 1 - \frac{{\dot{V}}_{\sec}}{{\dot{V}}_{in,g}} Separation at wall due to exceeding the loading limit in main stream .. math:: {\overline{z}}_{e} = \frac{u_{e}u_{con}}{{\overline{r}}_{z}} .. math:: d_{main,l}^{*} = \sqrt{w_{50}\frac{18\eta_{visc}}{\left( \rho_{s} - \rho_{g} \right){\overline{z}}_{e}}} .. math:: k = \begin{cases} 0.81 & \mu_{in} < 2.2 \cdot 10^{- 5} \\ 0.15 + 0.66\exp\left( - \left( \frac{\mu_{in} - 2.2 \cdot 10^{- 5}}{0.015 - 2.2 \cdot 10^{- 5}} \right)^{0.6} \right) & 2.2 \cdot 10^{- 5} \leq \mu_{in} < 0.015 \\ 0.15 + 0.66\exp\left( - \left( \frac{0.1 - 0.015}{0.1 - \mu_{in}} \right)^{0.1}\left( \frac{\mu_{in}}{0.015} \right)^{0.6} \right) & 0.015 \leq \mu_{in} \leq 0.1 \\ 0.15 & \mu_{in} > 0.1 \\ \end{cases} .. math:: \mu_{main} = K_{main}\left( \frac{d_{main,l}^{*}}{d_{50}} \right)\left( 10\mu_{in} \right)^{k} .. math:: \eta_{main,l} = 1 - \frac{\mu_{main}}{\mu_{in}} Separation in the internal vortex of main stream .. math:: d_{main,v}^{*} = \sqrt{\frac{18\eta_{visc}0.9{\dot{V}}_{in,g}}{\left( \rho_{s} - \rho_{g} \right)u_{f}^{2}2\pi h_{sep}}} .. math:: \eta_{main,v}(d) = \begin{cases} 0 & \frac{d}{d_{main,v}^{*}} < D^{- 1} \\ 0.5\left\{ 1 + \cos\left\lbrack 0.5\pi\left( 1 - \frac{\log\left( \frac{d}{d_{main,v}^{*}} \right)}{\log D} \right) \right\rbrack\ \right\} & D^{- 1} \leq \frac{d}{d_{main,v}^{*}} \leq D \\ 1 & \frac{d}{d_{main,v}^{*}} > D \\ \end{cases} Separation at wall due to exceeding the loading limit in secondary stream .. math:: \mu_{\sec} = \begin{cases} 6\mu_{main} & \mu_{in} \geq 6\mu_{main} \\ \mu_{in} & \mu_{in} < 6\mu_{main} \\ \end{cases} .. math:: \eta_{sec,l} = 1 - \frac{\mu_{\sec}}{\mu_{in}} Separation at vortex finder of secondary stream .. math:: d_{sec,v}^{*} = \sqrt{\frac{18\eta_{visc}{\dot{V}}_{\sec}}{\left( \rho_{s} - \rho_{g} \right)\left( \frac{2}{3}u_{f} \right)^{2}2\pi h_{f}}} .. math:: \eta_{sec,v}(d) = \begin{cases} 0 & \frac{d}{d_{sec,v}^{*}} < D^{- 1} \\ 0.5\left\{ 1 + \cos\left\lbrack 0.5\pi\left( 1 - \frac{\log\left( \frac{d}{d_{sec,v}^{*}} \right)}{\log D} \right) \right\rbrack\ \right\} & D^{- 1} \leq \frac{d}{d_{sec,v}^{*}} \leq D \\ 1 & \frac{d}{d_{sec,v}^{*}} > D \\ \end{cases}, \text{with } D = 3 Overall separation .. math:: \eta_{main}(d) = \begin{cases} \eta_{main,l} + \left( 1 - \eta_{main,l} \right)\eta_{main,v}(d) & \mu_{in} > \mu_{main} \\ \eta_{main,v}(d) & \mu_{in} \leq \mu_{main} \\ \end{cases} .. math:: \eta_{\sec}(d) = \begin{cases} \eta_{sec,l} + \left( 1 - \eta_{sec,l} \right)\eta_{sec,v}(d) & \mu_{in} > \mu_{\sec} \\ \eta_{sec,v}(d) & \mu_{in} \leq \mu_{\sec} \\ \end{cases} .. math:: \eta_{tot}(d) = \eta_{adj}\left( w_{split}\eta_{main}(d) + \left( 1 - w_{split} \right)\eta_{\sec}(d) \right) .. math:: {\dot{m}}_{s,out,s} = {\dot{m}}_{in,s}\sum_{d}^{}{R_{in}(d)\eta_{tot}(d)} .. math:: {\dot{m}}_{s,out,g} = 0 .. math:: {\dot{m}}_{g,out,s} = {\dot{m}}_{in,s}\left( 1 - \sum_{d}^{}{R_{in}(d)\eta_{tot}(d)} \right) .. math:: {\dot{m}}_{g,out,g} = {\dot{m}}_{in,g} .. note:: Notations: +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | Symbol | Units | Type | Description | +==============================+==================+======+=============================================================================================================================+ | :math:`\beta` | [-] | | Relative width of cyclone gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\delta` | [°] | UP | Angle of attack of blades in axial gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\varepsilon` | [°] | UP | Spiral angle in spiral gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\lambda_{0}` | [-] | UP | Wall friction coefficient of pure gas | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\lambda_{s}` | [-] | | Wall friction coefficient of solids-containing gas | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\mu_{in}` | [kg/kg] | | Solids loading at inlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\mu_{main}` | [kg/kg] | | Threshold for solids loading in main stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\mu_{\sec}` | [kg/kg] | | Threshold for solids loading in secondary stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\eta_{adj}` | [-] | UP | Separation efficiency adjustment factor | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\eta_{main}` | [-] | | Overall separation efficiency in main stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\eta_{main,l}` | [-] | | Separation efficiency due to exceeding of solids loading limit in main stream (from main stream to solids output) | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\eta_{main,v}` | [-] | | Separation efficiency in internal vortex (from internal vortex to solids output) | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\eta_{\sec}` | [-] | | Overall separation efficiency in secondary stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\eta_{sec,l}` | [-] | | Separation efficiency due to exceeding of solids loading limit in secondary stream (from secondary stream to solids output) | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\eta_{sec,v}` | [-] | | Separation efficiency at vortex finder (from vortex finder to solids output) | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\eta_{tot}` | [-] | | Total separation efficiency of cyclone | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\eta_{visc}` | [Pa s] | MDB | Dynamic viscosity of gas at inlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\rho_{g}` | [kg/m\ :sup:`3`] | MDB | Gas density at inlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`\rho_{s}` | [kg/m\ :sup:`3`] | MDB | Solids density at inlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`a` | [m] | | Height of blades channel in axial gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`A_{con}` | [m\ :sup:`2`] | | Lateral area of the conical part | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`A_{con/2}` | [m\ :sup:`2`] | | Lateral area of the top half of conical part | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`A_{cyl}` | [m\ :sup:`2`] | | Lateral area of the cylindrical part | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`A_{e1}` | [m\ :sup:`2`] | | Average wall area considered for the first revolution after entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`A_{f}` | [m\ :sup:`2`] | | Lateral area of vortex finder | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`A_{sed}` | [m\ :sup:`2`] | | Sedimentation area | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`A_{sp}` | [m\ :sup:`2`] | | Frictional area of the spiral in spiral gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`A_{top}` | [m\ :sup:`2`] | | Area of upper wall | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`A_{tot}` | [m\ :sup:`2`] | | Total wall friction area | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`b_{e}` | [m] | UP/ | Width of gas entry/blade channel | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`d` | [m] | SP | Particle diameter | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`d_{50}` | [m] | SP | Particle size median | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`d_{b}` | [m] | UP | Thickness of blades in axial gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`d_{exit}` | [m] | UP | Diameter of particles exit | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`d_{f}` | [m] | UP | Diameter of vortex finder | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`d_{o}` | [m] | UP | Outer diameter of cyclone | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`d_{main,l}^{*}` | [m] | | Cut size of separation on the first revolution due to exceeding the loading limit | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`d_{main,v}^{*}` | [m] | | Cut size of separation in internal vortex of main stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`d_{sec,v}^{*}` | [m] | | Cut size of separation at vortex finder in secondary stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`D` | [-] | UP | Coefficient for grid efficiency curve calculation according to Muschelknautz | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`h_{con}` | [m] | | Height of the cone part of cyclone | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`h_{con,eff}` | [m] | | Effective height of the cone part of cyclone | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`h_{cyl}` | [m] | UP | Height of the cylindrical part of cyclone | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`h_{e}` | [m] | UP | Height of gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`h_{f}` | [m] | UP | Height (depth) of vortex finder | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`h_{sep}` | [m] | | Height of separation zone | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`h_{tot}` | [m] | UP | Total height of cyclone | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`k` | [-] | | Exponent for solids loading threshold in main stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`K_{main}` | [-] | UP | Constant for solids loading threshold in main stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\dot{m}}_{in,g}` | [kg/s] | SP | Gas mass flow at inlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\dot{m}}_{in,s}` | [kg/s] | SP | Solids mass flow at inlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\dot{m}}_{out,s,s}` | [kg/s] | | Solids mass flow at solids outlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\dot{m}}_{out,s,g}` | [kg/s] | | Gas mass flow at solids outlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\dot{m}}_{out,g,s}` | [kg/s] | | Solids mass flow at gas outlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\dot{m}}_{out,g,g}` | [kg/s] | | Gas mass flow at gas outlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`n` | [-] | | Parameter for calculating secondary stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`N_{b}` | [#] | UP | Number of blades in axial gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\overline{r}}_{con}` | [m] | | Mean radius of the conical part | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`r_{core}` | [m] | UP | Core radius of blades in axial gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`r_{e}` | [m] | | Radius of the middle gas streamline at gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\overline{r}}_{e}` | [m] | | Mean radius of the gas streamline at gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`r_{exit}` | [m] | | Radius of the particles exit | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`r_{exit,eff}` | [m] | | Effective radius of the particles exit | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`r_{f}` | [m] | | Radius of vortex finder | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`r_{o}` | [m] | | Outer radius of cyclone | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\overline{r}}_{z}` | [m] | | Reference mean radius | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`R_{in}(d)` | [-] | | Mass fraction of particles with size :math:`d` at inlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`u_{con}` | [m/s] | | Tangential velocity at mean cone radius | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`u_{e}` | [m/s] | | Tangential velocity at gas streamline radius at gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`u_{f}` | [m/s] | | Tangential velocity at vortex finder | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`u_{o}` | [m/s] | | Tangential velocity at outer cyclone radius | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`v_{e}` | [m/s] | | Inlet velocity in the middle gas streamline at gas entry | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\dot{V}}_{in,g}` | [m\ :sup:`3`/s] | | Gas volume flow at inlet | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\dot{V}}_{\sec}` | [m\ :sup:`3`/s] | | Gas volume flow of secondary stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`w_{50}` | [m/s] | | Sinking speed at which 50% of particles are sedimented at wall | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`w_{split}` | [-] | | Fraction of material going to main stream | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ | :math:`{\overline{z}}_{e}` | [m\ :sup:`2`/s] | | Mean centrifugal acceleration along streamline | +------------------------------+------------------+------+-----------------------------------------------------------------------------------------------------------------------------+ - UP: User-defined model parameters - MDB: Value from materials database - SP: Value from the input stream .. note:: Model parameters: +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | Name | Symbol | Units | Description | Values | +=============+=====================+=======+==============================================================================+================================================+ | d_o | :math:`d_{o}` | [m] | Outer diameter of cyclone | ≥0.01 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | h_tot | :math:`h_{tot}` | [m] | Total height of cyclone | ≥0.01 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | h_cyl | :math:`h_{cyl}` | [m] | Height of the cylindrical part of cyclone | ≥0.01 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | d_f | :math:`d_{f}` | [m] | Diameter of vortex finder | ≥0.01 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | h_f | :math:`h_{f}` | [m] | Height (depth) of vortex finder | ≥0.01 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | d_exit | :math:`d_{exit}` | [m] | Diameter of particle exit | ≥0.01 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | Entry shape | | | Gas entry shape | Rectangular slot/Full spiral/Half spiral/Axial | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | b_e | :math:`b_{e}` | [m] | Width of gas entry | ≥0.01 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | h_e | :math:`h_{e}` | [m] | Height of gas entry | ≥0.01 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | epsilon | :math:`\varepsilon` | [°] | Spiral angle in spiral gas entry | [0...360] | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | N_b | :math:`N_{b}` | [#] | Number of blades in axial gas entry | ≥1 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | d_b | :math:`d_{b}` | [m] | Thickness of blades in axial gas entry | ≥0 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | r_core | :math:`r_{core}` | [m] | Core radius of blades in axial entry | ≥0 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | Blade shape | | | Blades shapes in axial gas entry | Simple straight/Curved/Curved and twisted | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | delta | :math:`\delta` | [°] | Angle of attack of blades in axial gas entry | [15...30] | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | lambda_0 | :math:`\lambda_{0}` | [-] | Wall friction coefficient of pure gas | ≥0 | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | D | :math:`D` | [-] | Coefficient for grid efficiency curve calculation according to Muschelknautz | [2...4] | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | K_main | :math:`K_{main}` | [-] | Constant for solids loading threshold in main stream | [0.02...0.03] | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | eta_adj | :math:`\eta_{adj}` | [-] | Separation efficiency adjustment factor | [0...1] | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ | Plot | | | Whether to generate plots | YES/NO | +-------------+---------------------+-------+------------------------------------------------------------------------------+------------------------------------------------+ .. seealso:: - Muschelknautz, U. (2019). L3.4 Zyklone zum Abscheiden fester Partikel aus Gasen. In: Stephan, P., Kabelac, S., Kind, M., Mewes, D., Schaber, K., Wetzel, T. (eds) VDI-Wärmeatlas. Springer Reference Technik. Springer Vieweg, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-52989-8_91 - A demostration file at ``Example Flowsheets/Units/Cyclone Muschelknautz.dlfw``. |