# Cyclone

Solids-gas separation according to Muschelknautz

Constant geometric parameters

r_{o} = 0.5d_{o}

r_{f} = 0.5d_{f}

r_{exit} = 0.5d_{exit}

b_{e} = \begin{cases} \text{user-defined} & \text{rect slot, full/half spiral entry} \\ r_{o} - r_{core} & \text{axial entry} \end{cases}

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}

{\overline{r}}_{con} = 0.5\left( r_{o} + r_{exit} \right)

r_{exit,eff} = \begin{cases} r_{f} & r_{exit} \leq r_{f} \\ r_{exit} & r_{exit} > r_{f} \\ \end{cases}

\beta = \frac{b_{e}}{r_{o}}

h_{con} = h_{tot} - h_{cyl}

h_{con,eff} = \left( \frac{r_{o} - r_{exit,eff}}{r_{o} - r_{exit}} \right)h_{con}

h_{sep} = h_{cyl} + h_{con,eff} - h_{f}

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}

A_{cyl} = 2\pi r_{o}h_{cyl}

A_{con} = \pi\left( r_{o} + r_{exit,eff} \right)\sqrt{h_{con,eff}^{2} + \left( r_{o} - r_{exit,eff} \right)^{2}}

A_{top} = \pi r_{o}^{2} - \pi r_{f}^{2}

A_{f} = 2\pi r_{f}h_{f}

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}

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}}

A_{sed} = A_{cyl} + A_{con/2}

A_{e1} = \frac{2\pi r_{o}h_{e}}{2}

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

{\dot{V}}_{in,g} = \frac{{\dot{m}}_{in,g}}{\rho_{g}}

\mu_{in} = \frac{{\dot{m}}_{in,s}}{{\dot{m}}_{in,g}}

\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}

\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

{\overline{r}}_{e} = r_{o} - \frac{\alpha b_{e}}{2}

{\overline{r}}_{z} = \sqrt{{\overline{r}}_{e}{\overline{r}}_{con}}

Velocities

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}

w_{50} = \frac{0.5\left( 0.9{\dot{V}}_{in,g} \right)}{A_{sed}}

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}

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}}}}

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}}}}

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

n = \frac{\ln\frac{u_{f}}{u_{o}}}{\ln\frac{r_{o}}{r_{f}\ }}

{\dot{V}}_{\sec} = {\dot{V}}_{in,g}\left( 0.0497 + 0.0684n + 0.0949n^{2} \right)

w_{split} = 1 - \frac{{\dot{V}}_{\sec}}{{\dot{V}}_{in,g}}

Separation at wall due to exceeding the loading limit in main stream

{\overline{z}}_{e} = \frac{u_{e}u_{con}}{{\overline{r}}_{z}}

d_{main,l}^{*} = \sqrt{w_{50}\frac{18\eta_{visc}}{\left( \rho_{s} - \rho_{g} \right){\overline{z}}_{e}}}

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}

\mu_{main} = K_{main}\left( \frac{d_{main,l}^{*}}{d_{50}} \right)\left( 10\mu_{in} \right)^{k}

\eta_{main,l} = 1 - \frac{\mu_{main}}{\mu_{in}}

Separation in the internal vortex of main stream

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}}}

\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

\mu_{\sec} = \begin{cases} 6\mu_{main} & \mu_{in} \geq 6\mu_{main} \\ \mu_{in} & \mu_{in} < 6\mu_{main} \\ \end{cases}

\eta_{sec,l} = 1 - \frac{\mu_{\sec}}{\mu_{in}}

Separation at vortex finder of secondary stream

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}}}

\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

\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}

\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}

\eta_{tot}(d) = \eta_{adj}\left( w_{split}\eta_{main}(d) + \left( 1 - w_{split} \right)\eta_{\sec}(d) \right)

{\dot{m}}_{s,out,s} = {\dot{m}}_{in,s}\sum_{d}^{}{R_{in}(d)\eta_{tot}(d)}

{\dot{m}}_{s,out,g} = 0

{\dot{m}}_{g,out,s} = {\dot{m}}_{in,s}\left( 1 - \sum_{d}^{}{R_{in}(d)\eta_{tot}(d)} \right)

{\dot{m}}_{g,out,g} = {\dot{m}}_{in,g}

Note

Notations:

Symbol | Units | Type | Description
--- | --- | --- | ---
\beta | [-] |  | Relative width of cyclone gas entry
\delta | [°] | UP | Angle of attack of blades in axial gas entry
\varepsilon | [°] | UP | Spiral angle in spiral gas entry
\lambda_{0} | [-] | UP | Wall friction coefficient of pure gas
\lambda_{s} | [-] |  | Wall friction coefficient of solids-containing gas
\mu_{in} | [kg/kg] |  | Solids loading at inlet
\mu_{main} | [kg/kg] |  | Threshold for solids loading in main stream
\mu_{\sec} | [kg/kg] |  | Threshold for solids loading in secondary stream
\eta_{adj} | [-] | UP | Separation efficiency adjustment factor
\eta_{main} | [-] |  | Overall separation efficiency in main stream
\eta_{main,l} | [-] |  | Separation efficiency due to exceeding of solids loading limit in main stream (from main stream to solids output)
\eta_{main,v} | [-] |  | Separation efficiency in internal vortex (from internal vortex to solids output)
\eta_{\sec} | [-] |  | Overall separation efficiency in secondary stream
\eta_{sec,l} | [-] |  | Separation efficiency due to exceeding of solids loading limit in secondary stream (from secondary stream to solids output)
\eta_{sec,v} | [-] |  | Separation efficiency at vortex finder (from vortex finder to solids output)
\eta_{tot} | [-] |  | Total separation efficiency of cyclone
\eta_{visc} | [Pa s] | MDB | Dynamic viscosity of gas at inlet
\rho_{g} | [kg/m3] | MDB | Gas density at inlet
\rho_{s} | [kg/m3] | MDB | Solids density at inlet
a | [m] |  | Height of blades channel in axial gas entry
A_{con} | [m2] |  | Lateral area of the conical part
A_{con/2} | [m2] |  | Lateral area of the top half of conical part
A_{cyl} | [m2] |  | Lateral area of the cylindrical part
A_{e1} | [m2] |  | Average wall area considered for the first revolution after entry
A_{f} | [m2] |  | Lateral area of vortex finder
A_{sed} | [m2] |  | Sedimentation area
A_{sp} | [m2] |  | Frictional area of the spiral in spiral gas entry
A_{top} | [m2] |  | Area of upper wall
A_{tot} | [m2] |  | Total wall friction area
b_{e} | [m] | UP/ | Width of gas entry/blade channel
d | [m] | SP | Particle diameter
d_{50} | [m] | SP | Particle size median
d_{b} | [m] | UP | Thickness of blades in axial gas entry
d_{exit} | [m] | UP | Diameter of particles exit
d_{f} | [m] | UP | Diameter of vortex finder
d_{o} | [m] | UP | Outer diameter of cyclone
d_{main,l}^{*} | [m] |  | Cut size of separation on the first revolution due to exceeding the loading limit
d_{main,v}^{*} | [m] |  | Cut size of separation in internal vortex of main stream
d_{sec,v}^{*} | [m] |  | Cut size of separation at vortex finder in secondary stream
D | [-] | UP | Coefficient for grid efficiency curve calculation according to Muschelknautz
h_{con} | [m] |  | Height of the cone part of cyclone
h_{con,eff} | [m] |  | Effective height of the cone part of cyclone
h_{cyl} | [m] | UP | Height of the cylindrical part of cyclone
h_{e} | [m] | UP | Height of gas entry
h_{f} | [m] | UP | Height (depth) of vortex finder
h_{sep} | [m] |  | Height of separation zone
h_{tot} | [m] | UP | Total height of cyclone
k | [-] |  | Exponent for solids loading threshold in main stream
K_{main} | [-] | UP | Constant for solids loading threshold in main stream
{\dot{m}}_{in,g} | [kg/s] | SP | Gas mass flow at inlet
{\dot{m}}_{in,s} | [kg/s] | SP | Solids mass flow at inlet
{\dot{m}}_{out,s,s} | [kg/s] |  | Solids mass flow at solids outlet
{\dot{m}}_{out,s,g} | [kg/s] |  | Gas mass flow at solids outlet
{\dot{m}}_{out,g,s} | [kg/s] |  | Solids mass flow at gas outlet
{\dot{m}}_{out,g,g} | [kg/s] |  | Gas mass flow at gas outlet
n | [-] |  | Parameter for calculating secondary stream
N_{b} | [#] | UP | Number of blades in axial gas entry
{\overline{r}}_{con} | [m] |  | Mean radius of the conical part
r_{core} | [m] | UP | Core radius of blades in axial gas entry
r_{e} | [m] |  | Radius of the middle gas streamline at gas entry
{\overline{r}}_{e} | [m] |  | Mean radius of the gas streamline at gas entry
r_{exit} | [m] |  | Radius of the particles exit
r_{exit,eff} | [m] |  | Effective radius of the particles exit
r_{f} | [m] |  | Radius of vortex finder
r_{o} | [m] |  | Outer radius of cyclone
{\overline{r}}_{z} | [m] |  | Reference mean radius
R_{in}(d) | [-] |  | Mass fraction of particles with size d at inlet
u_{con} | [m/s] |  | Tangential velocity at mean cone radius
u_{e} | [m/s] |  | Tangential velocity at gas streamline radius at gas entry
u_{f} | [m/s] |  | Tangential velocity at vortex finder
u_{o} | [m/s] |  | Tangential velocity at outer cyclone radius
v_{e} | [m/s] |  | Inlet velocity in the middle gas streamline at gas entry
{\dot{V}}_{in,g} | [m3/s] |  | Gas volume flow at inlet
{\dot{V}}_{\sec} | [m3/s] |  | Gas volume flow of secondary stream
w_{50} | [m/s] |  | Sinking speed at which 50% of particles are sedimented at wall
w_{split} | [-] |  | Fraction of material going to main stream
{\overline{z}}_{e} | [m2/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 | d_{o} | [m] | Outer diameter of cyclone | ≥0.01
h_tot | h_{tot} | [m] | Total height of cyclone | ≥0.01
h_cyl | h_{cyl} | [m] | Height of the cylindrical part of cyclone | ≥0.01
d_f | d_{f} | [m] | Diameter of vortex finder | ≥0.01
h_f | h_{f} | [m] | Height (depth) of vortex finder | ≥0.01
d_exit | d_{exit} | [m] | Diameter of particle exit | ≥0.01
Entry shape |  |  | Gas entry shape | Rectangular slot/Full spiral/Half spiral/Axial
b_e | b_{e} | [m] | Width of gas entry | ≥0.01
h_e | h_{e} | [m] | Height of gas entry | ≥0.01
epsilon | \varepsilon | [°] | Spiral angle in spiral gas entry | [0…360]
N_b | N_{b} | [#] | Number of blades in axial gas entry | ≥1
d_b | d_{b} | [m] | Thickness of blades in axial gas entry | ≥0
r_core | 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 | \delta | [°] | Angle of attack of blades in axial gas entry | [15…30]
lambda_0 | \lambda_{0} | [-] | Wall friction coefficient of pure gas | ≥0
D | D | [-] | Coefficient for grid efficiency curve calculation according to Muschelknautz | [2…4]
K_main | K_{main} | [-] | Constant for solids loading threshold in main stream | [0.02…0.03]
eta_adj | \eta_{adj} | [-] | Separation efficiency adjustment factor | [0…1]
Plot |  |  | Whether to generate plots | YES/NO

See also

- 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`.