# Unit library

An operation unit performs one of the following tasks:

• Streams processing: examples are Inlet flow, Outlet flow, Mixer and Splitter.

• Steady-state simulation: example is Screen.

• Dynamic simulation: examples are Granulator, Agglomerator and Bunker.

## Inlet flow

This unit allows defining the parameters of the input material, showing in the figure below.

## Outlet flow

This unit serves to connect the output material flows, as shown in the figure below.

## Mixer

A mixer mixes two input material streams ($$In1$$, $$In2$$) into the one output stream ($$Out$$), as shown in the scheme below.

The output stream will be defined for all time points for which the streams $$In1$$ and $$In2$$ are defined. Mixing of more streams can be implemented by connecting of several mixers sequentially.

The parameters of output stream are calculated as:

\begin{align}\begin{aligned}\dot{m}_{out} &= \dot{m}_{in1} + \dot{m}_{in2}\\\dot{H}_{out} &= \dot{H}_{in1} + \dot{H}_{in2}\\T_{out} &= f(h_{out}) = f \left( \frac{\dot{H}_{out}}{\dot{m}_{out}} \right)\\P_{out} &= min( P_{in1},P_{in2} )\end{aligned}\end{align}

Note

Notations:

$$\dot{m}$$ – mass flow

$$\dot{H}$$ - enthalpy flow

$$h$$ - specific enthalpy

$$T$$ - temperature

$$P$$ - pressure

All secondary attributes of output stream, such as phase fractions, compounds fractions and multidimensional distributions are calculated depending on mass fractions of input streams.

a demostration file at Example Flowsheets/Units/Mixer.dlfw.

## Splitter

A splitter divides input stream ($$In$$) into two output streams ($$Out1$$ and $$Out2$$), as shown in the figure below.

Both output streams are defined for the same set of time points for which the input stream has been defined. The splitting of input stream into more than two fractions can be done by sequential connection of several splitter units.

You can specify the splitting factor $$K_{splitt}$$, which is defined in following equations. Here $$\dot{m}$$ is a mass flow.

\begin{align}\begin{aligned}\dot{m}_{out1} &= K_{splitt} \cdot \dot{m}_{in}\\\dot{m}_{out2} &= (1-K_{splitt} ) \cdot \dot{m}_{in}\end{aligned}\end{align}

Note

Notations:

$$\dot{m}$$ - mass flow

$$K_{splitt}$$ - Splitting factor

Note

Input parameters needed for the simulation:

Name

Description

Units

Boundaries

Ksplitt

Splitting factor

[–]

0 ≤ Ksplitt ≤ 1

a demostration file at Example Flowsheets/Units/Splitter.dlfw.

## Screen

Screen unit is designed for classification of input material into two fractions according to particle size distribution (PSD), as shown below.

In Dyssol, 4 models are available to describe the screen grade efficiency:

• Plitt’s model

• Molerus & Hoffmann model

• Probability model

• Teipel / Hennig model

In the following figure, several grade efficiency curves for different parameters of separations sharpness are shown.

Note

This figure only applies to the Plitt’s model and Molerus & Hoffmann model.

### Plitt’s model

This model is described using the following equation:

$G(x_i) = 1 - exp\left(-0.693\,\left(\frac{x_i}{x_{cut}}\right)^\alpha\right)$

Note

Notations applied in the models:

$$G(x_i)$$ – grade efficiency: mass fraction of material within the size class $$i$$ in the feed ($$\dot{m}_{i,input}$$) that leaves the screen in the coarse stream ($$\dot{m}_{i,coarse}$$)

$$x_{cut}$$ – cut size of the classification model in meter

$$\alpha$$ – sharpness of separation

$$x_i$$ – size of a particle

Note

Input parameters needed for the simulation:

Name

Symbol

Description

Units

Boundaries

Xcut

$$x_{cut}$$

Cut size of the classification model

[m]

Xcut > 0

Alpha

$$\alpha$$

Sharpness of separation

[–]

0 ≤ Alpha ≤ 100

a demostration file at Example Flowsheets/Units/Screen Plitt.dlfw.

Plitt, L.R.: The analysis of solid–solid separations in classifiers. CIM Bulletin 64 (708), p. 42–47, 1971.

### Molerus & Hoffmann model

This model is described using the following equation:

$G(x_i) = \dfrac{1}{1 + \left( \dfrac{x_{cut}}{x_i} \right)^2 \cdot exp\left( \alpha \,\left( 1 - \left(\dfrac{x_i}{x_{cut}}\right)^2 \right)\right)}$

Note

Notations applied in the models:

$$G(x_i)$$ – grade efficiency: mass fraction of material within the size class $$i$$ in the feed that leaves the screen in the coarse stream

$$x_{cut}$$ – cut size of the classification model

$$\alpha$$ – sharpness of separation

$$x_i$$ – size of a particle

Note

Input parameters needed for the simulation:

Name

Symbol

Description

Units

Boundaries

Xcut

$$x_{cut}$$

Cut size of the classification model

[m]

Xcut > 0

Alpha

$$\alpha$$

Sharpness of separation

[–]

0 < Alpha ≤ 100

a demostration file at Example Flowsheets/Units/Screen Molerus-Hoffmann.dlfw.

Molerus, O.; Hoffmann, H.: Darstellung von Windsichtertrennkurven durch ein stochastisches Modell, Chemie Ingenieur Technik, 41 (5+6), 1969, pp. 340-344.

### Probability model

This model is described using the following equation:

$G(x_i) = \dfrac{ \sum\limits^{x_i}_{0} e^{-\dfrac{(x_i - \mu)^2}{2\sigma^2}} }{ \sum\limits^{N}_{0} e^{-\dfrac{(x_i - \mu)^2}{2\sigma^2}} }$

Note

Notations applied in this model:

$$G(x_i)$$ – grade efficiency: mass fraction of material within the size class $$i$$ in the feed that leaves the screen in the coarse stream

$$x_i$$ – size of a particle

$$\sigma$$ – standard deviation of the normal output distribution

$$\mu$$ – mean of the normal output distribution

$$N$$ – number of classes of particle size distribution

Note

Input parameters needed for the simulation:

Name

Symbol

Description

Units

Boundaries

Mean

$$\mu$$

Mean of the normal output distribution

[m]

Mean > 0

Standard deviation

$$\sigma$$

Standard deviation of the normal output distribution

[m]

Standard deviation > 0

a demostration file at Example Flowsheets/Units/Screen Probability.dlfw.

Radichkov, R.; Müller, T.; Kienle, A.; Heinrich, S.; Peglow, M.; Mörl, L.: A numerical bifurcation analysis of continuous fluidized bed spray granulation with external product classification, Chemical Engineering and Processing 45, 2006, pp. 826–837.

### Teipel / Hennig model

This model is described using the following equation:

$G(x_i) = \left( 1- \left( 1 + 3 \cdot \left( \dfrac{x_i}{x_{cut}} \right)^{\left(\dfrac{x_i}{x_{cut}} + \alpha \right)\cdot \beta} \right)^{-1/2} \right) \cdot (1 - a) + a$

Note

Notations applied in the models:

$$G(x_i)$$ – grade efficiency: mass fraction of material within the size class $$i$$ in the feed that leaves the screen in the coarse stream

$$x_{cut}$$ – cut size of the classification model

$$\alpha$$ – sharpness of separation

$$\beta$$ - sharpness of separation

$$a$$ - separation offset

$$x_i$$ – size of a particle

Note

Input parameters needed for the simulation:

Name

Symbol

Description

Units

Boundaries

Xcut

$$x_{cut}$$

Cut size of the classification model

[m]

Xcut > 0

Alpha

$$\alpha$$

Sharpness of separation 1

[–]

0 < Alpha ≤ 100

Beta

$$\beta$$

Sharpness of separation 2

[–]

0 < Beta ≤ 100

Offset

$$a$$

Separation offset

[–]

0 ≤ Offset ≤ 1

a demostration file at Example Flowsheets/Units/Screen Teipel-Hennig.dlfw.

Hennig, M. and Teipel, U. (2016), Stationäre Siebklassierung. Chemie Ingenieur Technik, 88: 911–918.

## Crusher

A crusher comminutes the input material stream and reduces the average particle size. The schema is illustrated below.

This unit can be described using 3 models in Dyssol:

• Bond’s model

• Cone model

• Const model

### Bond’s model

This model is used to perform milling of the input stream. The crushing is performed according to the model proposed by Bond. The simplification is made, and the particle size distribution of the output stream is described by the normal function.

$x_{80,out} = \dfrac{1}{ \left( \dfrac{P}{10\,w_i\,\dot{m}} + \dfrac{1}{\sqrt{x_{80,in}}} \right)^2}$
$\mu = x_{80,out} - 0.83\sigma$
$q_3(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-\dfrac{(x-\mu)^2}{2\sigma^2}}$

Note

Notations applied in this model:

$$x_{80,out}$$ – characteristic particle size of the output stream

$$x_{80,in}$$ – characteristic particle size of the input stream

$$w_i$$ – Bond Work Index, dependent on the material

$$P$$ – power input

$$\dot{m}$$ – mass flow of solids in the input stream

$$q_3(x)$$ – output mass related density distribution

$$\sigma$$ – standard deviation of the output normal distribution

$$\mu$$ – mean value of the output normal distribution

Note

Solid phase and particle size distribution are required for the simulation.

Note

Input parameters needed for the simulation:

Name

Symbol

Description

Units

Boundaries

P

$$P$$

Power input

[kW]

P > 0

Wi

$$w_i$$

Bond work index

[kWh/t]

1 ≤ Wi ≤ 100

Standard deviation

$$\sigma$$

Standard deviation of the output distribution

[m]

Standard deviation > 0

a demostration file at Example Flowsheets/Units/Crusher Bond.dlfw.

1. F.C. Bond, Crushing and grinding calculation – Part I, British Chemical Engineering 6 (6) (1961) 378-385.

2. F.C. Bond, Crushing and grinding calculation – Part II, British Chemical Engineering 6 (8), (1961) 543-548.

3. Denver Sala Basic: Selection Guide for Process Equipment, 1993.

#### Average Bond Work Indices for various materials

Material

Work Bond Index [kWh/t]

Material

Work Bond Index [kWh/t]

Andesite

20.08

Iron ore, oolitic

12.46

Barite

5.2

Iron ore, taconite

16.07

Basalt

18.18

13.09

Bauxite

9.66

12.02

14.8

Limestone

14

Clay

6.93

Manganese ore

13.42

Coal

14.3

Magnesite

12.24

Coke

16.84

Molybdenum

14.08

Copper ore

13.99

Nickel ore

15.02

Diorite

22.99

Oil shale

17.43

Dolomite

12.4

Phosphate rock

10.91

Emery

62.45

Potash ore

8.86

Feldspar

11.88

Pyrite ore

9.83

Ferro-chrome

8.4

Pyrrhotite ore

10.53

Ferro-manganese

9.13

Quartzite

10.54

Ferro-silicon

11

Quartz

14.93

Flint

28.78

Rutile ore

13.95

Fluorspar

9.8

Shale

17.46

Gabbro

20.3

Silica sand

15.51

Glass

13.54

Silicon carbide

27.46

Gneiss

22.14

Slag

11.26

Gold ore

16.42

Slate

15.73

Granite

16.64

Sodium silicate

14.74

Graphite

47.92

Spodumene ore

11.41

Gravel

17.67

Syenite

14.44

Gypsum rock

7.4

Tin ore

11.99

Iron ore ,hematite

14.12

Titanium ore

13.56

Iron ore, hematite-specular

15.22

Trap rock

21.25

Iron ore, magnetite

10.97

Zinc ore

12.72

### Cone model

The model is described below as

$w_{out,i} = \sum\limits^{i}_{k=0} w_{in,k} \cdot S_k \cdot B_{ki} + (1-S_i)\,w_{in,i}$

Note

Notations:

$$w_{out,i}$$ – mass fraction of particles with size $$i$$ in output distribution

$$w_{in,i}$$ – mass fraction of particles with size $$i$$ in inlet distribution

$$S_k$$ – mass fraction of particles with size $$k$$, which will be crushed

$$B_{ki}$$ – mass fraction of particles with size $$i$$, which get size after breakage less or equal to $$k$$

$$S_k$$ is described by the King selection function.

$\begin{split} S_k = \begin{cases} 0 & x_k \leqslant x_{min} \\ 1 - \dfrac{x_{max} - x_i}{x_{max} - x_{min}} & x_{min} < x_k < x_{max} \\ 1 & x_k \geqslant x_{max} \end{cases}\end{split}$
\begin{align}\begin{aligned}x_{min} = CSS \cdot \alpha_1\\x_{max} = CSS \cdot \alpha_2\end{aligned}\end{align}

Note

Notations:

$$x_k$$ – mean particle diameter in size-class $$k$$

$$CSS$$ – close size setting of a cone crusher

$$\alpha_1, \alpha_2, n$$ – parameters of the King selection function

$$B_{ki}$$ is calculated by the Vogel breakage function.

$\begin{split}B_{ki} = \begin{cases} 0.5\, \left( \dfrac{x_i}{x_k} \right)^q \cdot \left( 1 + \tanh \left( \dfrac{x_k - x'}{x'} \right) \right) & i \geqslant k \\ 0 & i < k \end{cases}\end{split}$

Note

Notations:

$$x'$$ – minimum fragment size which can be achieved by crushing

$$q$$ – parameter of the Vogel breakage function

Note

Solid phase and particle size distribution are required for the simulation.

Note

Input parameters needed for the simulation:

Name

Symbol

Description

Units

Boundaries

CSS

$$CSS$$

Close size setting of a cone crusher. Parameter of the King selection function

[m]

CSS > 0

alpha1

$$\alpha_1$$

Parameter of the King selection function

[–]

0.5 ≤ alpha1 ≤ 0.95

alpha2

$$\alpha_2$$

Parameter of the King selection function

[–]

1.7 ≤ alpha2 ≤ 3.5

n

$$n$$

Parameter of the King selection function

[–]

1 ≤ n ≤ 3

d’

$$x'$$

Minimum fragment size achieved by crushing. Parameter of the Vogel breakage function

[m]

d’ > 0

q

$$q$$

Parameter of the Vogel breakage function

[–]

a demostration file at Example Flowsheets/Units/Crusher Cone.dlfw.

1. King, R. P., Modeling and simulation of mineral processing systems, Butterworth & Heinemann, Oxford, 2001.

2. Vogel, L., Peukert, W., Modelling of Grinding in an Air Classifier Mill Based on A Fundamental Material Function, KONA, 21, 2003, 109-120.

### Const output model

This model sets a normal distribution with the specified constant parameters to the output stream. Outlet distribution does not depend on the inlet distribution.

$q_3(x) = \frac{1}{\sigma\sqrt{2\pi}}\,e^{-\dfrac{(x-\mu)^2}{2\sigma^2}}$

Note

Notations:

$$q_3(x)$$ – output mass related density distribution

$$\sigma$$ – standard deviation of the output normal distribution

$$\mu$$ – mean value of the output normal distribution

Note

Solid phase and particle size distribution are required for the simulation.

Note

Input parameters needed for the simulation:

Name

Symbol

Description

Units

Boundaries

Mean

$$\mu$$

Mean of the normal output distribution

[m]

Mean > 0

Standard deviation

$$\sigma$$

Standard deviation of the normal output distribution

[m]

Standard deviation > 0

a demostration file at Example Flowsheets/Units/Crusher Const.dlfw.

## Bunker

Bunker unit performs accumulation of the solid part of the input material with ideal mixing, see figure below.

The model takes into account only the solid phase, the rest of the phases are bypassed.

$\frac{dm}{dt} = \dot{m}_{in} - \dot{m}_{out}$

Two models for the bunker outflow $$m_{out}$$ are available:

• Adaptive: User defines only the target mass $$m_{target}$$ of the bunker and $$\dot{m}_{out}$$ is being adjusted by the system to match the user-defined target mass $$m_{target}$$, depending on inflow mass $$\dot{m}_{in}$$, current bunker mass $${m}$$ and $$m_{target}$$: .

$\dot{m}_{out} = \dot{m}_{in}\left(\frac{2m}{m + m_{target}}\right)^2$
• Constant: User defines timepoints with the desired bunker outflow $$\dot{m}_{requested}$$. The system tries to provide this outflow, if enough material $${m}$$ is in bunker. Otherwise the $$\dot{m}_{out} = \dot{m}_{in}$$. The smoothing function is implemented to let the numerical solver provide reliable results:

$f_{smooth} = \frac{1}{2} + \frac{1}{2} \cdot \tanh{\left(50\cdot\left(m - \dot{m}_{requested}\cdot{dt}\right)\right)}$
$\dot{m}_{out} = f_{smooth} \cdot \dot{m}_{requested} + \left(1 - f_{smooth} \right) \cdot \min\left(\dot{m}_{in}, \dot{m}_{requested}\right)$

To correctly take into account the dynamics of the process, norms of each overall parameter (mass flow, temperature, pressure) are maintained as:

$\frac{d||X||}{dt} = (X(t) - X(t-1))^2 - ||X||$

For compounds fractions:

$\frac{d||C||}{dt} = \sqrt{\sum_{i}^{N_{c}}{(w_{i}(t) - w_{i}(t-1))^2}} - ||C||$

For each distributed parameter:

$\frac{d||D_{i}||}{dt} = \sqrt{\sum_{j}^{N_{D_{i}}}{(w_{i,j}(t) - w_{i,j}(t-1))^2}} - ||D||$

Note

Notations:

$${m}$$ – current mass inside the bunker

$$m_{target}$$ – target mass inside the bunker

$$\dot{m}_{in}$$ – solids input mass flow

$$\dot{m}_{out}$$ – solids output mass flow

$$X(t)$$ – value of an overall parameter at time point $$t$$

$$w(t)$$ – mass fraction at time point $$t$$

$$N_{c}$$ – number of defined compounds

$$N_{D_{i}}$$ – number of classes in distribution $$i$$

Note

Solid phase is required for the simulation.

Note

Input parameters needed for the simulation:

Name

Description

Units

Boundaries

Target mass

Target mass within the bunker

[kg]

Target mass > 0

Relative tolerance

Relative tolerance for DAE solver

[-]

>0 (0 for flowsheet-wide value)

Absolute tolerance

Absolute tolerance for DAE solver

[-]

>0 (0 for flowsheet-wide value)

a demostration file at Example Flowsheets/Units/Bunker.dlfw.

## Granulator

This unit represents a simplified model of a fluidized bed granulation reactor. The model does not take into account attrition of particles inside the apparatus and does not keep properly any secondary distributed properties except size.

### Continuous granulator

$\frac{dq_{3,i}}{dt} = -G_e\,\frac{q_{3,i} - q_{3,i-1}\,\left(\frac{d_{p,i}}{d_{p,i-1}}\right)^3}{\Delta d_i} + \frac{\dot{m}_{in}}{M_{tot}}\,q_{3,i}^{in} - \frac{\dot{m}_{out}}{M_{tot}}\,q_{3,i}$
$G_e = \frac{2\dot{m}_e}{\rho_{s,susp} \cdot A_{tot}}$
$A_{tot} = \frac{6M_{tot}}{\rho_s} \sum\limits_{i} \frac{q_{3,i}\cdot \Delta d_i}{d_{p,i}}$
$\dot{m}_e = \dot{m}_{s,susp}\,(1 - K_{os})$
$\dot{m}_{out} = \dot{m}_{in} + \dot{m}_{e}$
$\dot{m}_{dust} = \dot{m}_{s,susp}\cdot K_{os} + (\dot{m}_{susp} - \dot{m}_{s,susp} + \dot{m}_{fl,g})$

### Batch granulator

$\frac{d(M_{tot}q_{3,i})}{dt} = -G_e\,\frac{M_{tot}q_{3,i} - M_{tot}q_{3,i-1}\,\left(\frac{d_{p,i}}{d_{p,i-1}}\right)^3}{\Delta d_i}$
$G_e = \frac{2\dot{m}_{s,susp}}{\rho_{s,susp} \cdot A_{tot}}$
$A_{tot} = \frac{6M_{tot}}{\rho_s} \sum\limits_{i} \frac{q_{3,i}\cdot \Delta d_i}{d_{p,i}}$
$\frac{dM_{tot}}{dt} = \dot{m}_{s,susp}$
$\dot{m}_{exh} = \dot{m}_{l,susp} + \dot{m}_{fl,gas}$

Note

Notations:

$$q_3$$ – mass density distribution of particles inside apparatus

$$q_3^{in}$$ – mass density distribution of external particles from ExternalNuclei stream

$$\Delta d$$ – class size

$$d_p$$ – particle diameter in a class

$$\dot{m}_{in}$$ – mass flow of input nuclei

$$\dot{m}_{out}$$ – output mass flow of the product

$$\dot{m}_{dust}$$ – output mass flow from the DustOutput

$$\dot{m}_{susp}$$ – total mass flow of the suspension

$$\dot{m}_{s,susp}$$ – mass flow of the solid phase in the Suspension inlet

$$\dot{m}_{fl,g}$$ – mass flow of the gas phase in the FluidizationGas inlet

$$\dot{m}_{exh}$$ – output mass flow from the ExhaustGasOutput

$$\dot{m}_{e}$$ – effective mass stream of the injected suspension

$$M_{tot}$$ – holdup mass

$$\rho_{s,susp}$$ – density of solids in the holdup

$$G_{e}$$ – effective growth rate

$$A_{tot}$$ – total surface of particles in the granulator

$$K_{os}$$ – overspray part in the suspension

Note

particle size distribution is required for the simulation. This unit is applied for solid, liquid and gas phases.

Note

Input parameters needed for the simulation:

Name

Symbol

Description

Units

Boundaries

Kos

$$K_{os}$$

Overspray part in the suspension

[–]

0 ≤ Kos ≤ 1

RTol

Relative tolerance for equation solver

[–]

0 < RTol ≤ 1

ATol

Absolute tolerance for equation solver

[–]

0 < ATol ≤ 1

Note

State variables:

Name

Symbol

Description

Units

Atot

$$A_{tot}$$

Total surface of particles in the granulator

[$$m^2$$]

Mtot

$$M_{tot}$$

Total mass of all particles in the granulator

[kg]

Mout

$$\dot{m}_{out}$$

Output mass flow of the product

[kg/s]

Mdust

$$\dot{m}_{dust}$$

Output mass flow of dust

[kg/s]

G

$$G_{e}$$

Effective growth rate

[m/s]

PSDi

$$q_{3,i}$$

Mass density distribution of particles

[1/m]

a demostration file at Example Flowsheets/Units/Granulator.dlfw.

S.Heinrich, M. Peglow, M. Ihlow, M. Henneberg, L. Mörl, Analysis of the start-up process in continuous fluidized bed spray granulation by population balance modelling, Chem. Eng. Sci. 57 (2002) 4369-4390.

## Agglomerator

This unit represents a simplified model of agglomeration process, see figure below.

The model does not take into account attrition of particles inside the apparatus and does not keep properly any secondary distributed property except size.

Mass related density distribution of output stream is calculated according to following equations:

$\frac{\partial n(v,t)}{\partial t} = B_{agg}(n,v,t) - D_{agg}(n,v,t) + \dot{n}_{in}(t) - \dot{n}_{out}(t)$
$B_{agg}(n,v,t) = \frac{1}{2}\,\beta_0\,\textstyle \int\limits_{0}^{v} \beta(u,v - u)\,n(u,t)\,n(v-u,t)\,du$
$D_{agg}(n,v,t) = \beta_0\,n(v,t)\, \textstyle \int\limits_{0}^{\infty}\,\beta(v,u)\,n(u,t)\,du$
$\dot{m}_{out}(t) = \dot{m}_{in}(t)$

Note

Notations:

$$v,u$$ – volumes of agglomerating particles

$$n(v,t)$$ – number density function

$$\dot{n}_{in}(t)$$, $$\dot{n}_{out}(t)$$ – number density functions of inlet and outlet streams, correspondingly

$$B_{agg}(n,v,t)$$, $$D_{agg}(n,v,t)$$ – birth and death rates of particles with volume $$v$$ caused due to agglomeration

$$\beta_0$$ – agglomeration rate constant, dependent on operating conditions but independent from particle sizes

$$\beta(v,u)$$ – the agglomeration kernel, see section Kernels.

$$t$$ – time

$$\dot{m}_{in}$$ – mass flow in the input stream

$$\dot{m}_{out}$$ – mass flow in the output stream

Note

solid phase and particle size distribution are required for the simulation.

The method of calculating $$B_{agg}(n,v,t)$$ and $$D_{agg}(n,v,t)$$ is determined by the selected solver via unit parameter Agglomeration solvers.

Note

Input parameters needed for the simulation:

Name

Symbol

Description

Units

Boundaries

Beta0

$$\beta_0$$

Size independent agglomeration rate constant

[–]

0 < Beta0 ≤ $$10^{20}$$

Step

Maximum time step of internal DAE solver. Default value is 0.

[–]

0 ≤ Step ≤ $$10^{9}$$

Solver

Solver used to calculate birth and death rates

[–]

Kernel

Agglomeration kernel type, must be an integer

[–]

0 ≤ Kernel ≤ 9

Rank

Rank of the kernel (applied for FFT solver only), must be an integer

[–]

1 ≤ Rank ≤ 10

a demostration file at Example Flowsheets/Units/Agglomerator.dlfw.

V.Skorych, M. Dosta, E.-U. Hartge, S. Heinrich, R. Ahrens, S. Le Borne, Investigation of an FFT-based solver applied to dynamic flowsheet simulation of agglomeration processes, Advanced Powder Technology 30 (3) (2019), 555-564.

### Kernels

The agglomeration kernels are applied to describe the agglomeration frequency between particles of volumes $$v$$ and $$u$$, which produce a new particle with the size $$(v + u)$$. In Dyssol environment, 10 types of kernels are numbered with integers from 0 to 9, as listed below.

Number

Name

Kernel equation

0

Constant

$$\beta (u,v)=1$$

1

Sum

$$\beta (u,v)=u+v$$

2

Product

$$\beta (u,v)=uv$$

3

Brownian

$$\beta (u,v)=\left(u^{\frac{1}{3}}+v^{\frac{1}{3}} \right)\,\left(u^{-\frac{1}{3}}+v^{-\frac{1}{3}} \right)$$

4

Shear

$$\beta (u,v)=\left(u^{\frac{1}{3}}+v^{\frac{1}{3}} \right)^{\frac{7}{3}}$$

5

Peglow

$$\beta (u,v)=\dfrac{ (u+v)^{0.71} }{(uv)^{0.062} }$$

6

Coagulation

$$\beta(u,v)=u^{\frac{2}{3}}+v^{\frac{2}{3}}$$

7

Gravitational

$$\beta(u,v)=\left(u^{\frac{1}{3}}+v^{\frac{1}{3}} \right)^2 \left|u^{\frac{1}{6}}-v^{\frac{1}{6}} \right|$$

8

Kinetic energy

$$\beta(u,v)=\left(u^{\frac{1}{3}}+v^{\frac{1}{3}} \right)^2 \, \sqrt{\frac{1}{u}+\frac{1}{v}}$$

9

Thompson

$$\beta(u,v)=\dfrac{(u-v)^2}{u+v}$$

## Time delay

Constant delay of input signal

### Simple shift

Copies all time points $$t$$ from the input stream $$In$$ to the output stream $$Out$$ at the timepoint $$t + \Delta t$$, delaying the signal by a constant value $$\Delta t$$.

### Norm-based

$\frac{dm}{dt} = \dot{m}_{in}(t-\Delta t) - m$

To correctly take into account the dynamics of the process, norms of each overall parameter (mass flow, temperature, pressure) are maintained as:

$\frac{d||X||}{dt} = (X(t) - X(t-1))^2 - ||X||$

For phase fractions:

$\frac{d||P||}{dt} = \sqrt{\sum_{i}^{N_{P}}{(w_{i}(t) - w_{i}(t-1))^2}} - ||P||$

For compound fractions in each phase:

$\frac{d||C_{i}||}{dt} = \sqrt{\sum_{j}^{N_{C_{i}}}{(w_{i,j}(t) - w_{i, j}(t-1))^2}} - ||C||$

For each distributed parameter:

$\frac{d||D_{i}||}{dt} = \sqrt{\sum_{j}^{N_{D_{i}}}{(w_{i,j}(t) - w_{i,j}(t-1))^2}} - ||D||$

Note

Notations:

$${m}$$ – current mass

$$\dot{m}_{in}$$ – input mass flow

$$\Delta t$$ – time delay

$$X(t)$$ – value of an overall parameter at time point $$t$$

$$w(t)$$ – mass fraction at time point $$t$$

$$N_{P}$$ – number of defined phases

$$N_{C_{i}}$$ – number of defined compounds in phase $$i$$

$$N_{D_{i}}$$ – number of classes in distribution $$i$$

Note

Model parameters:

Name

Symbol

Description

Units

Boundaries

Time delay

Model to use

Norm based, Simple shift

Time delay

$$\Delta t$$

Time delay

[s]

>=0

Relative tolerance

Relative tolerance for DAE solver

[-]

>0 (0 for flowsheet-wide value)

Absolute tolerance

Absolute tolerance for DAE solver

[-]

>0 (0 for flowsheet-wide value)

a demostration file at Example Flowsheets/Units/Time Delay.dlfw.

## Cyclone

Solids-gas separation according to Muschelknautz

Constant geometric parameters

$r_{o} = 0.5d_{o}$
$r_{f} = 0.5d_{f}$
$r_{exit} = 0.5d_{exit}$
$\begin{split}b_{e} = \begin{cases} \text{user-defined} & \text{rect slot, full/half spiral entry} \\ r_{o} - r_{core} & \text{axial entry} \end{cases}\end{split}$
$\begin{split}r_{e} = \begin{cases} r_{o} - 0.5b_{e} & \text{rect slot, axial entry} \\ r_{o} + 0.5b_{e} & \text{full spiral entry} \\ r_{o} & \text{half spiral entry} \\ \end{cases}\end{split}$
${\overline{r}}_{con} = 0.5\left( r_{o} + r_{exit} \right)$
$\begin{split}r_{exit,eff} = \begin{cases} r_{f} & r_{exit} \leq r_{f} \\ r_{exit} & r_{exit} > r_{f} \\ \end{cases}\end{split}$
$\beta = \frac{b_{e}}{r_{o}}$
$h_{con} = h_{tot} - h_{cyl}$
$h_{con,eff} = \left( \frac{r_{o} - r_{exit,eff}}{r_{o} - r_{exit}} \right)h_{con}$
$h_{sep} = h_{cyl} + h_{con,eff} - h_{f}$
$\begin{split}a = \begin{cases} \text{-} & \text{rect slot, full/half spiral entry} \\ \sin(\delta)\frac{\pi\left( r_{o} + r_{core} \right)}{N_{b}} - d_{b} & \text{axial entry} \\ \end{cases}\end{split}$
$A_{cyl} = 2\pi r_{o}h_{cyl}$
$A_{con} = \pi\left( r_{o} + r_{exit,eff} \right)\sqrt{h_{con,eff}^{2} + \left( r_{o} - r_{exit,eff} \right)^{2}}$
$A_{top} = \pi r_{o}^{2} - \pi r_{f}^{2}$
$A_{f} = 2\pi r_{f}h_{f}$
$\begin{split}A_{tot} = \begin{cases} A_{cyl} + A_{con} + A_{f} + A_{top} & \text{rect slot, axial entry} \\ A_{cyl} + A_{con} + A_{f} + A_{top} - \varepsilon r_{o}h_{e} & \text{full/half spiral entry} \\ \end{cases}\end{split}$
$A_{con/2} = \pi\left( r_{o} + {\overline{r}}_{con} \right)\sqrt{\left( \frac{h_{con}}{2} \right)^{2} + \left( r_{o} - {\overline{r}}_{con} \right)^{2}}$
$A_{sed} = A_{cyl} + A_{con/2}$
$A_{e1} = \frac{2\pi r_{o}h_{e}}{2}$
$\begin{split}A_{sp} = \begin{cases} \text{-} & \text{rect slot, axial entry} \\ \varepsilon\left( \frac{b + 2r_{o}}{2}\left( b_{e} + h_{e} \right) \right) & \text{full spiral entry} \\ \varepsilon r_{o}\left( b_{e} + h_{e} \right) & \text{half spiral entry} \\ \end{cases}\end{split}$

Operational parameters

${\dot{V}}_{in,g} = \frac{{\dot{m}}_{in,g}}{\rho_{g}}$
$\mu_{in} = \frac{{\dot{m}}_{in,s}}{{\dot{m}}_{in,g}}$
$\begin{split}\lambda_{s} = \begin{cases} \lambda_{0}\left( 1 + 2\sqrt{\mu_{in}} \right) & \mu_{in} \leq 1 \\ \lambda_{0}\left( 1 + 3\sqrt{\mu_{in}} \right) & \mu_{in} > 1 \\ \end{cases}\end{split}$
$\begin{split}\alpha = \begin{cases} \frac{1}{\beta}\left( 1 - \sqrt{1 + 4\left\lbrack \left( \frac{\beta}{2} \right)^{2} - \left( \frac{\beta}{2} \right) \right\rbrack\sqrt{1 - \frac{1 - \beta^{2}}{1 + \mu_{in}}\left( 2\beta - \beta^{2} \right)}} \right) & \text{rect slot, full/half spiral entry} \\ \begin{cases} 0.85 & \text{simple straight blades} \\ 0.95 & \text{curved blades} \\ 1.05 & \text{curved and twisted blades} \\ \end{cases} & \text{axial entry} \\ \end{cases}\end{split}$

Geometric parameters

${\overline{r}}_{e} = r_{o} - \frac{\alpha b_{e}}{2}$
${\overline{r}}_{z} = \sqrt{{\overline{r}}_{e}{\overline{r}}_{con}}$

Velocities

$\begin{split}v_{e} = \begin{cases} {\dot{V}}_{in,g}/\left( b_{e}h_{e} \right) & \text{rect slot, full/half spiral entry} \\ {\dot{V}}_{in,g}/\left( ab_{e}N_{b} \right) & \text{axial entry} \\ \end{cases}\end{split}$
$w_{50} = \frac{0.5\left( 0.9{\dot{V}}_{in,g} \right)}{A_{sed}}$
$\begin{split}u_{o} = \begin{cases} \frac{v_{e}\frac{r_{e}}{r_{o}}}{\alpha} & \text{rect slot entry} \\ \frac{v_{e}\frac{r_{e}}{r_{o}}}{1 + \frac{\lambda_{s}}{2}\frac{A_{sp}}{{\dot{V}}_{in,g}}v_{e}\sqrt{\frac{r_{e}}{r_{o}}}\ } & \text{full/half spiral entry} \\ \frac{v_{e}\cos(\delta)\frac{r_{e}}{r_{o}}}{\alpha} & \text{axial entry} \\ \end{cases}\end{split}$
$u_{f} = \frac{u_{o}\frac{r_{o}}{r_{f}}}{1 + \frac{\lambda_{s}}{2}\frac{A_{tot}}{{\dot{V}}_{in,g}}u_{o}\sqrt{\frac{r_{o}}{r_{f}}}}$
$u_{e} = \frac{u_{o}\frac{r_{o}}{{\overline{r}}_{e}}\ }{1 + \frac{\lambda_{s}}{2}\frac{A_{e1}}{0.9{\dot{V}}_{in,g}}u_{o}\sqrt{\frac{r_{o}}{{\overline{r}}_{e}}}}$
$u_{con} = \frac{u_{o}\frac{r_{o}}{{\overline{r}}_{con}}\ }{1 + \frac{\lambda_{s}}{2}\frac{A_{sed}}{0.9{\dot{V}}_{in,g}}u_{o}\sqrt{\frac{r_{o}}{{\overline{r}}_{con}}}}$

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

${\overline{z}}_{e} = \frac{u_{e}u_{con}}{{\overline{r}}_{z}}$
$d_{main,l}^{*} = \sqrt{w_{50}\frac{18\eta_{visc}}{\left( \rho_{s} - \rho_{g} \right){\overline{z}}_{e}}}$
$\begin{split}k = \begin{cases} 0.81 & \mu_{in} < 2.2 \cdot 10^{- 5} \\ 0.15 + 0.66\exp\left( - \left( \frac{\mu_{in} - 2.2 \cdot 10^{- 5}}{0.015 - 2.2 \cdot 10^{- 5}} \right)^{0.6} \right) & 2.2 \cdot 10^{- 5} \leq \mu_{in} < 0.015 \\ 0.15 + 0.66\exp\left( - \left( \frac{0.1 - 0.015}{0.1 - \mu_{in}} \right)^{0.1}\left( \frac{\mu_{in}}{0.015} \right)^{0.6} \right) & 0.015 \leq \mu_{in} \leq 0.1 \\ 0.15 & \mu_{in} > 0.1 \\ \end{cases}\end{split}$
$\mu_{main} = K_{main}\left( \frac{d_{main,l}^{*}}{d_{50}} \right)\left( 10\mu_{in} \right)^{k}$
$\eta_{main,l} = 1 - \frac{\mu_{main}}{\mu_{in}}$

Separation in the internal vortex of main stream

$d_{main,v}^{*} = \sqrt{\frac{18\eta_{visc}0.9{\dot{V}}_{in,g}}{\left( \rho_{s} - \rho_{g} \right)u_{f}^{2}2\pi h_{sep}}}$
$\begin{split}\eta_{main,v}(d) = \begin{cases} 0 & \frac{d}{d_{main,v}^{*}} < D^{- 1} \\ 0.5\left\{ 1 + \cos\left\lbrack 0.5\pi\left( 1 - \frac{\log\left( \frac{d}{d_{main,v}^{*}} \right)}{\log D} \right) \right\rbrack\ \right\} & D^{- 1} \leq \frac{d}{d_{main,v}^{*}} \leq D \\ 1 & \frac{d}{d_{main,v}^{*}} > D \\ \end{cases}\end{split}$

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

$\begin{split}\mu_{\sec} = \begin{cases} 6\mu_{main} & \mu_{in} \geq 6\mu_{main} \\ \mu_{in} & \mu_{in} < 6\mu_{main} \\ \end{cases}\end{split}$
$\eta_{sec,l} = 1 - \frac{\mu_{\sec}}{\mu_{in}}$

Separation at vortex finder of secondary stream

$d_{sec,v}^{*} = \sqrt{\frac{18\eta_{visc}{\dot{V}}_{\sec}}{\left( \rho_{s} - \rho_{g} \right)\left( \frac{2}{3}u_{f} \right)^{2}2\pi h_{f}}}$
$\begin{split}\eta_{sec,v}(d) = \begin{cases} 0 & \frac{d}{d_{sec,v}^{*}} < D^{- 1} \\ 0.5\left\{ 1 + \cos\left\lbrack 0.5\pi\left( 1 - \frac{\log\left( \frac{d}{d_{sec,v}^{*}} \right)}{\log D} \right) \right\rbrack\ \right\} & D^{- 1} \leq \frac{d}{d_{sec,v}^{*}} \leq D \\ 1 & \frac{d}{d_{sec,v}^{*}} > D \\ \end{cases}, \text{with } D = 3\end{split}$

Overall separation

$\begin{split}\eta_{main}(d) = \begin{cases} \eta_{main,l} + \left( 1 - \eta_{main,l} \right)\eta_{main,v}(d) & \mu_{in} > \mu_{main} \\ \eta_{main,v}(d) & \mu_{in} \leq \mu_{main} \\ \end{cases}\end{split}$
$\begin{split}\eta_{\sec}(d) = \begin{cases} \eta_{sec,l} + \left( 1 - \eta_{sec,l} \right)\eta_{sec,v}(d) & \mu_{in} > \mu_{\sec} \\ \eta_{sec,v}(d) & \mu_{in} \leq \mu_{\sec} \\ \end{cases}\end{split}$
$\eta_{tot}(d) = \eta_{adj}\left( w_{split}\eta_{main}(d) + \left( 1 - w_{split} \right)\eta_{\sec}(d) \right)$
${\dot{m}}_{s,out,s} = {\dot{m}}_{in,s}\sum_{d}^{}{R_{in}(d)\eta_{tot}(d)}$
${\dot{m}}_{s,out,g} = 0$
${\dot{m}}_{g,out,s} = {\dot{m}}_{in,s}\left( 1 - \sum_{d}^{}{R_{in}(d)\eta_{tot}(d)} \right)$
${\dot{m}}_{g,out,g} = {\dot{m}}_{in,g}$

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]

$$\mu_{main}$$

[kg/kg]

$$\mu_{\sec}$$

[kg/kg]

$$\eta_{adj}$$

[-]

UP

$$\eta_{main}$$

[-]

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/

$$d$$

[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

$$k$$

[-]

$$K_{main}$$

[-]

UP

$${\dot{m}}_{in,g}$$

[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

$$r_{e}$$

[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]

$$r_{exit,eff}$$

[m]

Effective radius of the particles exit

$$r_{f}$$

[m]

$$r_{o}$$

[m]

$${\overline{r}}_{z}$$

[m]

$$R_{in}(d)$$

[-]

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]

≥0

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

$$K_{main}$$

[-]

[0.02…0.03]

$$\eta_{adj}$$

[-]

[0…1]

Plot

Whether to generate plots

YES/NO

• A demostration file at Example Flowsheets/Units/Cyclone Muschelknautz.dlfw.