Template:Solids PSD method theory

User Defined Partition Curve

In the case of the partition curve, the model will distribute the feed material based on the user defined partition curve. The user defines the screen partition curve as the fraction of the feed to the screen reporting to the over size product. The screen model will ensure that the products follow this curve.

Whiten Method

This method is based on a model proposed by Whiten. The user may either specify:

• The d₅₀ of the screen; or
• The Screen aperture and Efficiency.

The Reduced Efficiency curve to the oversize is given in equation (1):

(1) [math]\displaystyle{ \mathbf{\mathit{E_{o a i}=\cfrac{exp(\alpha x_i)-1}{exp(\alpha x_i)+exp(\alpha)-2}}} }[/math]

where

[math]\displaystyle{ \mathbf{\mathit{x_i = \cfrac{Particle Diameter_i}{d_{50}}}} }[/math]

Particle Diameter_i = geometric mean of the size interval i.

d₅₀ = cut-size or separation size, the size which divides equally between oversize and undersize.

alpha(α) = efficiency parameter. High values of α (>9) indicate good separations. (The value of α will change the slope of the screen efficiency, or partition curve.)

If the user specifies the Screen aperture and efficiency, then the model calculates the theoretical d₅₀ using the following equation:

(2) [math]\displaystyle{ \mathbf{\mathit{d_{50}=\cfrac{\alpha A}{ln \left[\left(\cfrac{100}{100-E}-1\right)exp(\alpha)+\left(\cfrac{100}{100-E}\right)-2\right]}}} }[/math]

where

A = Nominal screen aperture.

E = Screen Efficiency at aperture size, typically 95%.

Fines Bypass

The mass of solid material reporting to the over size product may be influenced by the amount of liquid reporting to the over size. The fine material often bypasses to the over size with the liquid, or by adhering to the coarse material.

The corrected recovery to over size is calculated using equation (3):

(3) [math]\displaystyle{ \mathbf{\mathit{E_{o c i}=\cfrac{E_{o a i}-R_f}{1-R_f}}} }[/math]

where:

E_oci = corrected recovery to the over size

E_oai = actual recovery to the over size (calculated in equation (1) above)

R_f = Proportion of feed liquid reporting to the over size product.
Note: The R_f is only used to adjust the solids fine fraction to the screen oversize and NOT to set the fraction of liquid reporting to the oversize stream.

The Fraction of feed liquor reporting to the under size, C, can be found using equation (4):

(4) C = 1 - R_f

Whiten Beta Method

This is the modified method by Whiten to accommodate some abnormal "bumps" in the fine size end of the Efficiency curve.

The Reduced Efficiency curve to the oversize is then as follows:

(5) [math]\displaystyle{ \mathbf{\mathit{E_{o a i}=C\left[\cfrac{(1+\beta\beta^*x_i)(exp(\alpha)-1)}{exp(\alpha\beta^*x_i)+exp(\alpha)-2}\right]}} }[/math]

where:

Beta (β) - is the term introduced to control the initial rise in the curve at fine sizes. If this term is set to 0 then the equation is the same form as (1) in the previous heading.

BetaStar (β*) is derived iteratively from β so that E_oai = (1/2)C when d_i = d₅₀

References:

Napier-Munn et al "Mineral Comminution Circuits - Their Operation and Optimisation", Chapter 12, which references:

Whiten W.J. "Lecture notes for winter school on mineral processing." Dept. Min & Eng, University of Queensland, Aug 1966 (Lynch and Bull)

Whiten W.J. Private communication (JKMRC), 1996.

Karra Method

This method is based on a model proposed by V.K Karra¹ and is only valid for screen cut apertures greater than 1mm. The solids split is calculated using the d₅₀ of the screen. This may either be defined by the user (when CalcMethod=d50) or calculated by the model (when CalcMethod=Area) using the physical screen dimensions and feed size distribution.

With the Karra.CalcMethod set to 'd50' the user defines the cut point of the Screen, the d₅₀. The model then calculates the solids split directly, as shown in equation (8).

With the Karra.CalcMethod set to 'Area' the user defines:

1. The Screen area
2. The cut aperture (> 1mm - the model will not allow the user to specify a smaller aperture), and
3. Whether it is a wet screening application.

The model will then calculate the d₅₀ based on these user-defined parameters and on the size distribution of the feed. This d₅₀ is then used to calculate the solids split, as shown in equation (8).

The model calculates the d₅₀ of the Screen from the following equations¹

(6) [math]\displaystyle{ \mathbf{\mathit{d_{50} = h_T * Factor * \left[\cfrac{ \left( \cfrac{Theoretical Undersize (tph)}{Screen Area (m^2)} \right)} {ABCDEFG}\right]^{-0.148}}} }[/math]

where

• The Cut aperture h_T, in mm, is given by (7): [math]\displaystyle{ \mathbf{\mathit{h_T=\left(h+d\right)cos\varphi-d}} }[/math]

h - Aperture of square mesh, mm

d - Wire diameter, mm

[math]\displaystyle{ \mathbf{\mathit{\varphi}} }[/math] - Screen angle of inclination to the horizontal

• Factor is a tuning factor (default value is 1)
• the Theoretical Undersize is the mass of solids (in t/h) with sizes less than h_T

The modifying factors in the denominator of equation (6) are obtained as follows:

Factor A.	hT < 50.8mm	A = 12.1286 * (hT)0.3162 - 10.2991
	hT >= 50.8mm	A = 0.3388 * hT + 14.4122
Factor B.	Where Q - % Oversize in feed to screen deck.
	Q =< 87	B = 1.6 - 0.012 * Q
	Q =< 87	B obtained from values in Nordberg reference manual. (Previously from equation: B = 4.275 + 0.0425 * Q)
Factor C	Where R - % half size feed to the screen deck.
	R =< 30	C = 0.012 * R + 0.7
	30 < R < 55	C = 0.1528 (R)0.564
	55 =< R < 80	C = 0.0061 (R)1.37
	R >= 80	C = 0.05 * R - 1.5
Factor D.	Where S is deck location, top deck S = 1, second deck S = 2
		D = 1.1 - 0.1 * S
Factor E.	Wet Screening Factor, T = 1.26 * hT
	T < 1	E = 1.0
	1 =< T =< 2	E = T
	2 < T < 4	E = 1.5 + 0.25T
	4 =< T =< 6	E = 2.5
	6 < T =< 10	E = 3.25 - 0.125T
	10 < T < 12	E = 4.5 - 0.25T
	12 =< T =< 16	E = 2.1 - 0.05T
	16 < T < 24	E = 1.5 - 0.125T
	24 =< T =< 32	E = 1.35 - 0.00625T
	T > 32	E = 1.15
Factor F		F = U/1602, where U = Solids Density (kg/m2)
Factor G Near-Mesh factor		G = 0.844 * (1.0 - Xn/100)3.453, where Xn = % near size feed to the screen deck, % in the size interval 1.25hT to 0.75hT

Split Efficiency

The model then uses either the calculated or user defined d₅₀ of the screen to calculate the recovery to the over size in each size range (y_i) using the Rosin-Rammler equation with sharpness of 5.846:

(8) [math]\displaystyle{ \mathbf{\mathit{y_i = 1-exp\left(-0.693147(x_i)^{5.846}\right)}} }[/math]

where

[math]\displaystyle{ \mathbf{\mathit{x_i = \cfrac{Particle Diameter_i}{d_{50}}}} }[/math]

[math]\displaystyle{ Particle Diameter_i }[/math] is the geometric mean for the size interval

Notes:

• The Karra method is only valid for screen cut apertures greater than 1mm.
• When 'Area' is selected for Karra.CalcMethod the model is a form of "load based screen" and performance is a function of feed flowrate. Care should be taken, especially in steady state modelling, that the feed flowrate is representative of the expected operational flowrate.

Assumptions

1. The equations are based on screening crushed stone. While the characteristics of metallic ores are very similar, this may not be true for sand and gravel applications.
2. The Cut Aperture, of the screen deck is greater than 1mm.

Reference:

• V.K.Karra., "Development of a model for predicting the screening performance of a vibrating screen", CIM Bulletin, April 1979.

Rosin-Rammler Method

This method is based on a Rosin-Rammler type of function with the efficiency curve expression derived by Reid and Plitt.

The Efficiency curve to the oversize is given in equation (10):

(10) [math]\displaystyle{ \mathbf{\mathit{y_i' = 1-exp\left(-0.693147(x_i)^{m}\right)}} }[/math]

where

[math]\displaystyle{ \mathbf{\mathit{x_i = \cfrac{Particle Diameter_i}{d_{50}}}} }[/math]

Particle Diameter_i = geometric mean of the size interval i.

d₅₀ = cut-size or separation size, the size which divides equally between oversize and undersize.

m = sharpness parameter. High values of m for sharper separation.

References:

1. L.R.Plitt, A mathematical model of the hydrocyclone classifier, CIM Bulletin, December 1976
2. K.J. Reid, Derivation of an equation for classifier performance curves, Canadian Metallurgical Quarterly (1971)

Lynch Method

(11) Recovery to underflow on a corrected basis for the size interval

[math]\displaystyle{ \mathbf{\mathit{y_i'=\cfrac{exp\left(\alpha\cfrac{d_i}{d_{50}}\right)-1}{exp\left(\alpha\cfrac{d_i}{d_{50}}\right)+exp(\alpha)-2}}} }[/math]

where: d_i - geometric mean of the size interval

α = (1.54 * m) - 0.47

m = measure of the sharpness of separation

Note: The Lynch method is effectively the same as the simple Whiten method (without Beta), except alpha is calculated from m.

(12) The actual recovery to the underflow, y, is then calculated using the same equation for both of the above methods:

[math]\displaystyle{ \mathbf{\mathit{y=y'+R_f(1-y')}} }[/math]

where R_f = fraction of feed liquid reporting to the underflow product.

Reference:

L.R.Plitt, A mathematical model of the hydrocyclone classifier, CIM Bulletin, December 1976.

Del Villar and Finch Method

This method includes a term for the "fish hook" effect for entrainment.

(13) [math]\displaystyle{ \mathbf{\mathit{y_i = a_i+(1-a_i)(1-exp\left(-0.693147(x_i)^{m}\right)}} }[/math]

where

[math]\displaystyle{ \mathbf{\mathit{x_i = \cfrac{Particle Diameter_i}{d_{50}}}} }[/math]

[math]\displaystyle{ \mathit{a_i = R_f*\left(1-\cfrac{Particle Diameter_i}{d_{50}}\right)} }[/math] for [math]\displaystyle{ d_i \lt d_0 }[/math]

[math]\displaystyle{ \mathbf{\mathit{a_i = 0}} }[/math] for [math]\displaystyle{ d_i \gt d_0 }[/math]

ParticleDiameter_i = geometric mean of the size interval i.

d₅₀ = cut-size or separation size, the size which divides equally between oversize and undersize.

d₀ = largest particle size affected by the fish-hook for entrainment function.

R_f = Proportion of feed liquid reporting to the over size product.

m = sharpness parameter. High values of m for sharper separation.

Notes:

• At the finest particles the recovery approaches Rf. When Rf is equal to the liquid fraction to over size product then the finest particles are approaching the liquid split.
• If d₀ is less than or equal to the smallest size then this method is equivalent to the Rosin-Rammler Method.

References:

1. M.Frachon, J.J.Cilliers, A general model for hydrocyclone partition curves, Chemical Engineering Journal 73 (1999)
2. R.Del Villar, J.A.Finch, Modelling the cyclone performance with a size dependent entrainment factor, Minerals Engineering (1992)