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
.