View source for Tumbling Mill (Power, Morrell Discrete Shell)

== Description  ==

This article describes the '''Morrell Discrete Shell''' (Morrell D) method for estimating the power draw of a tumbling mill.{{Morrell_(1993)}}
The Morrel D model adopts a more sophisticated treatment of charge dynamics than the [[Tumbling Mill (Power, Morrell Continuum)|Morrell Continuum]] method:
* The total charge is subdivided into discrete layers, or shells, whose size and position may individually vary due to grinding media size, mill speed, lift bar geometry and other conditions.
* The rate at which potential and kinetic energy is imparted to each shell during mill rotation is calculated separately and subsequently summed to estimate the total mill power draw.

== Model theory ==

[[File:TumblingMillPower5.png|thumb|700px|thumb|Figure 1. Grate discharge tumbling mill profile illustrating the Morrell D model's charge shape and concentric shell layers (at increments of five for clarity).<br><br>The radial position (<math>r</math>), shoulder position (<math>\theta_{S25}</math>) and toe position (<math>\theta_{T25}</math>) of shell 25 are shown as an example.]]

[[File:TumblingMillDimensions2.png|thumb|525px|Figure 2. Schematic of a tumbling mill showing key dimensions.]]

This section summarises the main relations comprising Morrell's Discrete Shell tumbling mill power model. For a full description of the model algorithms, the reader is referred to Morrell's publicly available [http://dx.doi.org/10.13140/RG.2.1.3189.2248 dissertation].

Following the observation that ball and rock size has an effect on tumbling mill power draw, Morrell improved the [[Tumbling Mill (Power, Morrell Continuum)|Continuum]] model approach by treating the charge as a collection of individual rocks and balls, rather than a continuous phase. 

This was achieved by considering the charge as a number of layered, concentric annular sectors, or ''shells'', as shown in Figure 1. These shells slide against one another during mill rotation, leading to an increasing loss of rotational rate of between the layers due to slip as their distance from the outer mill shell increases.

The power imparted to each shell is therefore different, and the total power input to the motor of a tumbling mill is then:

:<math>\text{Gross power (kW)} = \text{No-load power} + \sum_{i=1}^{n}{P_r}</math>

where <math>P_r</math> is the power draw of discrete shell <math>i</math>, and <math>n</math> is the number of shells comprising the charge.

=== Power per shell ===

The power draw of an individual shell in the cylindrical section of the mill due to the potential energy of the charge, the kinetic energy of the charge and the potential energy of the slurry pool is:

:<math>
\begin{align}
P_r & = 2 \pi \int_{r - \bar x}^{r} \int_{\theta_{Tr}}^{\theta_{Sr}} N_r r^2 L \rho_c g \cos \theta \, {\rm d} \theta \, {\rm d} r + 4 \pi^3 \int_{r - \bar x}^{r} N_r^3 r^3 L \rho_c \, {\rm d}r + 2 \pi \int_{r - \bar x}^{r} \int_{\theta_{TO}}^{\theta_{Tr}} N_r R^2 L \rho_p g \cos \theta \, {\rm d} \theta \, {\rm d}r\\
& = \frac{2}{3} \pi LgN_r \left (r^3 - (r - \bar x)^3 \right ) \left ( \rho_c  (\sin \theta_{Sr} - \sin \theta_{Tr}) + \rho_p (\sin \theta_{Tr} - \sin \theta_{TO}) \right ) + \pi^3 L \rho_c N_r^3 \left ( r^4 - (r - \bar x)^4 \right )
\end{align}
</math>

where:
* <math>r</math> is the radial distance of the shell from the centre of rotation (m)
* <math>\bar x</math> is the thickness of the shell (m)
* <math>N_r</math> is the rotational rate at radial position <math>r</math> (revs/s)
* <math>L</math> is the mill length of cylindrical section (m)
* <math>\rho_c</math> is the  charge density (t/m<sup>3</sup>)
* <math>\rho_p</math> is the discharge pulp density (t/m<sup>3</sup>)
* <math>\theta_{Sr}</math> is the shoulder angle at radial position <math>r</math> (rad)
* <math>\theta_{Tr}</math> is the toe angle at radial position <math>r</math> (rad)
* <math>\theta_{TO}</math> is the slurry pool overflow angle (rad)
* <math>g</math> is acceleration due to gravity (m/s<sup>2</sup>)

and all angles are unit circle positive.

=== Shell rotational rate ===

Shells are assumed to slip against one another and progressively lose rotational speed as the radial position moves from the top of the lifters towards the centre of rotation.

The loss of rotational speed of a discrete shell, <math>\Delta \bar N</math>, at radial position <math>r</math> is:

:<math>\Delta \bar N = \frac{N_r z \gamma}{1 - z \left ( \frac{r_i}{r} \right )}</math>

where <math>r_i</math> is the radius of the inner surface of the shell, i.e. <math>r_i=r - \bar x</math>, and <math>\gamma</math> is a slip parameter which is expected to be related to the coefficient of friction.

The parameter <math>z</math> is an empirical term that relates the rotational rate of particles at radial positions within the charge to the rotational rate of the mill, and is defined as:

:<math>z = \left (  1 - J_{tr} \right )^{0.4532}</math>

where <math>J_{tr}</math> is the volume fraction mill filling at radial position <math>r</math>, i.e. the fraction of the total volume between the mill centre and shell radius that is filled with charge.

Shells located between the top of the lifters and outer mill shell are locked into place by the lifters, and rotate at the mill rotational speed without loss.

=== Characteristic media size ===

The thickness of each shell, <math>\bar x</math>, is related to the particle size distribution of the charge (balls and coarse rocks). Morrell adopted the concept of a ''characteristic media size'' to determine a representative layer thickness.

The characteristic media size is the number-weighted mean diameter of all particles from the coarsest in the charge to the largest that can fit within the grinding media interstices, the ''media size lower limit''. Therefore:

:<math>\bar x = \frac{\sum\limits_{i=1}^{j}n_ix_i}{\sum\limits_{i=1}^{j} n}</math>

where <math>n_i</math> is the number of particles in size interval <math>i</math>, for <math>i=1 \dots j</math>, with <math>i=1</math> being the coarsest fraction and <math>j</math> being the size fraction whose geometric mean size is the media size lower limit.

The number of particles in each size interval, <math>n_i</math>, is computed as:

:<math>n_i = \frac{6W_i}{\rho x_i^3 \pi}</math>

where <math>W_i</math> is the total mass of particles in size interval <math>i</math>, <math>\rho</math> is the density of particles, and <math>x_i</math> is the geometric mean size of size interval <math>i</math>.

The media size lower limit was established from measurement of the charge in a laboratory mill, in rough agreement with a theoretical analysis, and is defined as:

:<math> \text{Media size lower limit} =
    \begin{cases}
      0.133M_1 & \mbox{for ball mills}\\
      0.165M_2 & \mbox{for autogenous and semi-autogenous mills}\\
    \end{cases}
</math>

where:
* <math>M_1</math> is the geometric mean size of the coarsest size fraction in the mill, e.g. make-up ball diameter
* <math>M_2</math> is the 95% percent passing size (P<sub>95</sub>) of particles (balls or rocks) in the mill (or feed)

Calculation of the characteristic media size requires the full particle size distribution of balls and ore in the mill.

For ball mills, the characteristic media size may be estimated by neglecting ore particles and assuming the ball size size distribution is in an equilibrium state (ball make-up rate = ball ejection rate), in which case:

:<math>\bar x = 0.5559 D_B</math>

where <math>D_B</math> is the make-up ball diameter.

Note that Morrell's ball size equilibrium assumptions are also consistent with the tumbling mill media string approach described [[Tumbling Mill (Media Strings)|here]].

=== Charge position ===

Morrell developed the following series of relations to estimate the geometry of a shell at radial position <math>r</math>.

==== Toe and shoulder positions ====

The position of the '''toe''' of a shell, <math>\theta_{Tr}</math> (rad), is defined as: 

:<math>\theta_{Tr} = 2.5307 (1.2796 - J_{tr}) (1 - \rm e^{-9(\phi_c - \phi)}) + \frac{\pi}{2}</math>

where <math>\phi</math> (frac) is the theoretical [[Tumbling Mill (Speed)|fraction critical speed]], and the fraction of critical speed at which centrifuging actually occurs, <math>\phi_c</math> (frac), is:

:<math> \phi_c =
    \begin{cases}
      \phi & \phi>0.35(3.364-J_{tr})\\
      0.35(3.364-J_{tr})& \phi \leq 0.35(3.364-J_{tr})\\
    \end{cases}
</math>

The position of the '''shoulder''' of a shell, <math>\theta_{Sr}</math> (rad), is: 

:<math>\theta_{Sr} = \frac{\pi}{2} \big( (0.3386 + 0.1041 \phi) + (1.54 - 2.5673 \phi) J_{tr} \big) \left (\theta_{Tr} - \frac{\pi}{2} \right )</math>

This implementation of the Discrete Shell model further adopts the Delboni (2002) approach for shells at radial positions between the mill outer shell and the top of installed lifters.{{Delboni_and_Morrell_(2002)}}

The toe and shoulder positions of these outer shells are determined by applying Powell's (1991) method for computing the [[Tumbling Mill (Media Trajectory)|trajectory of grinding media elements]] in a tumbling mill.{{Powell_(1991)}}

The shoulder position of such shells is the point of departure of a media element of size <math>\bar x</math> from the tip of a lifter into free flight, from a starting position determined by the shell radial position, <math>r</math>. Similarly, the shell toe position is the point of impact with the outer mill shell of the same grinding media element in free flight .

The remaining shells at radial distances above the height of the lifters use Morrell's original shoulder and toe position relations described above.

==== Slurry pool position ====

The position of the '''slurry pool''' level, <math>\theta_{TO}</math> (rad), was assumed to be equal to the charge toe, <math>\theta_{T}</math>, for grate mills. For overflow mills, a value of 3.395 radians was calculated where the trunnion radius, <math>r_t</math>, is one quarter (0.25) of the mill radius, <math>r_m</math>.

This implementation of the Discrete Shell model improves estimation of the slurry pool position for overflow mills by adopting the approach described by Morrell (2016) and corrected by Shi (2016):{{Morrell_(2016)}}{{Shi_(2016)}}

:<math>\theta_{TO} = \arcsin \left ( \frac{r_t - h}{r_m} \right )</math>

where <math>h</math> is the height of slurry flow above the level of the trunnion lip: 

:<math>h = \frac{1}{0.67} \left ( \frac{2Q}{\pi r_t g^{0.5}} \right )^{\frac{2}{3}}</math>

and <math>Q</math> is the volumetric flow rate of pulp to/from the overflow mill.

==== Shell volume ====

The shell shoulder and toe positions only account for the active portion of the charge, i.e. the fraction not in free-flight between the should and the toe. The total volume of a shell, including the inactive portion, must be computed by the model algorithm to ensure the sum of all shell volumes equals the total charge volume. The total shell volume is therefore:

:<math>\text{Total shell volume} = \frac{\text{Active charge volume}}{\beta_r}</math>

where <math>\beta_r</math> is the fraction of total shell volume in the active portion. <math>\beta_r</math> is estimated as:

:<math>\beta_r = \frac{t_{cr}}{t_{cr}+t_{fr}}</math>

where <math>t_{cr}</math> is the time taken to travel between the toe and shoulder at radial position <math>r</math> within the active portion of the shell, and <math>t_{fr}</math> is the time taken to travel in free flight between the shoulder and toe at radial position <math>r</math>. i.e.,

:<math>t_{cr} = \frac{2 \pi + \theta+{Sr} - \theta_{Tr}}{2 \pi N_r}</math>

:<math>t_{fr} = 2r \left( \frac{\sin \theta_{Sr} - \sin \theta_{Tr}}{g} \right )^{0.5}</math>

=== Charge density ===

Morrell's 1993 Discrete Shell model algorithm assumed the grinding media interstices were completely filled with slurry. 

This implementation adopts Morrell's 1996 approach to compute charge density, which includes a term for the fraction of void space occupied by slurry:{{Morrell_(1996a)}}
:<math>\rho_c = \frac{J_t \rho_o (1 - E + EUS) + J_B( \rho_B - \rho_o)(1 - E) + J_tEU(1 - S)}{J_t}</math>

where:
* <math>J_t</math> is the volumetric fraction of the mill occupied by balls and coarse rock (including void space and interstitial slurry) (v/v).
* <math>\rho_o</math> is the density of ore (t/m<sup>3</sup>)
* <math>E</math> is volumetric fraction of interstitial void space in the charge, typically 0.4 (v/v)
* <math>U</math> is the volumetric fraction of interstitial grinding media voidage occupied by slurry (v/v), <math>U \leq 1</math>
* <math>S</math> is the volume fraction of solids in the mill discharge (v/v)
* <math>J_B</math> is the volumetric fraction of the mill occupied by balls (including voids) (v/v)

=== Conical ends ===

[[File:TumblingMillPower6.png|thumb|450px|Figure 3. Discretisation of cone end shells, Morrell Discrete Shell tumbling mill power model (after Morrell, 1993).{{Morrell_(1993)}}]]

The power draw of the cone end sections of the mill is approximated by further sub-dividing the discrete shells into vertical slices along the mill length dimension, as illustrated in Figure 3. 

The outer radial positions of the vertical slices are set to be consistent with the cone geometry. The shell thickness, <math>\bar x</math>, is also used as the slice width in the mill length direction.

Power in the cone slices is subsequently calculated using the same approach as the shells in the cylindrical section of the mill.

=== No-load power ===

This implementation of the Discrete Shell model uses Morrell's 1996 term for no-load power:<ref name="Morrell_(1996a)" />

:<math>\text{No-load power} = 1.68D^{2.05} \left [\phi (0.667L_d+L \right ]^{0.82}</math>

where <math>D</math> is mill diameter inside liners (m) and <math>L_d</math> is length of the cone end (m).

The length of the cone end, <math>L_d</math> (m), is:

:<math>L_d = (r_m - r_t) \tan \alpha_{c}</math>

where <math>\alpha_{c}</math> is the cone angle, measured as the angular displacement of the cone surface from the vertical direction.

=== Model algorithm ===

The Morrell D model adopts the algorithm outlined below to compute total mill power draw:

# The charge is divided into a number of shells based on the total charge volume, <math>J_t</math>, the user-defined characteristic media size, <math>\bar x</math>, and an estimate of the charge inner surface radius, <math>r_i</math>
# The radial position, rotational speed, power draw and volume of each individual shell is then computed using the relations described in the previous sections.
# The charge inner surface radius, and hence the number of shells, is re-estimated and the calculation steps 1 - 3 repeated until the sum of shell volumes matches the total charge volume.
# An inner loop around steps 1 - 2 also makes incremental adjustments to the characteristic media size value, <math>\bar x</math>, to minimise the numerical error associated with discretising the real number-valued charge volume into an integer number of shells with variable volume.
# Net power draw is calculated by summing the shell power values.
# The net power calculation process in steps 1 - 5 is conducted twice; once with the laboratory-observed slip parameter (<math>\gamma</math>) value of 0.02433, and then again with the industrial mill value of 0.0028 (or a user-specified value). This estimates the fractions of net power draw attributable to to ''attrition/abrasion'' (industrial mill only) and ''impact'' mechanisms (industrial and laboratory mills).
# Total power draw is finally calculated by adding the net power and no-load power values.

== Additional notes ==

The Morrell D model requires an estimation of the charge particle size distribution (balls and coarse ore), and lifter geometry, both of which are typically not available during normal mill operation. In addition, the model applies a a multi-step, looping computational algorithm which makes it unsuited to spreadsheet-style calculations. For these reasons, the Morrell D model has not been tested and applied as extensively as the Morrell Continuum and Empirical models.

Nevertheless, the Morrell D model may find utility when coupled with process models that predict mill load and ball particle size distributions (e.g. [[AG/SAG Mill (Mutambo)]] and [[Tumbling Mill (Media Strings)]]), and lifter wear measurements from modern laser scanning devices.

Like the Continuum and Empirical models, the Morrell D model is only valid for grate discharge mills that do not exhibit a slurry pool, i.e. <math>U \leq 1</math>. The slurry pool present in overflow discharge mills is included within the model formulation via the <math>\theta_{TO}</math> term, as described above.

== Excel ==

The Morrell Discrete Shell mill power model may be invoked from the Excel formula bar with the following function call:

<syntaxhighlight lang="vb">=mdMillPower_MorrellD(Parameters as Range)</syntaxhighlight>

{{Excel (Text, Help, No Arguments)}}

The ''Parameters'' array and model results are defined below in matrix notation, along with an example image showing the selection of the same arrays in the Excel interface:

{| 
|- style="vertical-align:top;"
|
:<math>Parameters=
\begin{bmatrix}
D\text{ (m)}\\
L\text{ (m)}\\
D_t\text{ (m)}\\
\alpha_{c}\text{ (degrees)}\\
h\text{ (m)}\\
\rho\text{ (rad)}\\
\text{Mill speed (rpm)}\\
J_t\text{ (v/v)}\\
J_B\text{ (v/v)}\\
U\text{ (v/v)}\\
\bar x\text{ (m)}\\
\rho_o\text{ (t/m}^{\text{3}}\text{)}\\
\rho_L\text{ (t/m}^{\text{3}}\text{)}\\
\rho_B\text{ (t/m}^{\text{3}}\text{)}\\
\text{Discharge pulp density (}\% \text{ w/w)}\\
\text{Discharge mechanism}\\
\gamma\\
Q_v\text{ (m}^{\text{3}}\text{/h)}\\
\end{bmatrix},\;\;\;\;\;\;

mdMillPower\_MorrellD=
\begin{bmatrix}

\begin{bmatrix}
\text{Gross power (kW)}\\
\text{No-loadpower (kW)}\\
\text{Net impact power (kW)}\\
\text{Net att/abr power (kW)}\\
\end{bmatrix}

\begin{bmatrix}
r_1 \text{ (m)}\\
\vdots\\
r_n \text{ (m)}
\end{bmatrix}

\begin{bmatrix}
\theta_{S1} \text{ (rad)}\\
\vdots\\
\theta_{Sn} \text{ (rad)}
\end{bmatrix}

\begin{bmatrix}
\theta_{T1} \text{ (rad)}\\
\vdots\\
\theta_{Tn} \text{ (rad)}
\end{bmatrix}

\begin{bmatrix}
P_1 \text{ (kW)}\\
\vdots\\
P_n \text{ (kW)}
\end{bmatrix}

\end{bmatrix}\;\;\;\;\;\;\;\;\;\;\;\;
</math>


where:
* <math>D_t</math> is the diameter of the discharge trunnion (m), i.e. <math>D_t = 2 r_t</math>
* <math>h</math> is the lifter height (m)
* <math>\rho</math> is the lifter face angle (deg.)
* <math>\text{Mill speed}</math> is the rotational speed of the mill (rpm)
* <math>\rho_L</math> is the density of liquids (t/m<sup>3</sup>)
* <math>\text{Discharge pulp density}</math> is the mass fraction of solids in the discharge pulp (% w/w)
* <math>\text{Discharge mechanism}</math> is the discharge configuration type, ''0 = Grate, 1 = Overflow''
* <math>Q_v</math> is the volumetric discharge flow rate of pulp from the mill (m<sup>3</sup>/h)
* <math>\text{Net impact power}</math> is the power draw attributable to the impact mechanism (kW)
* <math>\text{Net att/abr power}</math> is the power draw attributable to the attrition/abrasion mechanisms (kW)
* <math>n</math> is the number of discrete shells
* <math>r_i</math> is the radius of shell <math>i</math> (m)
* <math>\theta_{Si}</math> is the angular position of the shoulder of shell <math>i</math> (rad)
* <math>\theta_{Ti}</math> is the angular position of the toe of shell <math>i</math> (rad)
* <math>P_i</math> is the power drawn by shell <math>i</math> (kW)

| [[File:TumblingMillPower7.png|frame|Figure 4. Example showing the selection of the '''Parameters''' (blue frame), and '''Results''' (light blue frame) arrays in Excel.]]
|}

== SysCAD ==

The Morrell Discrete Shell power model is an optional calculation for tumbling mill units. If selected, the input and display parameters below are shown.

{{SysCAD_Table_Header}}
|-
! colspan="3" style="text-align:left;" |''MorrellC''
|-
|MillDiameter
|style="background: #eaecf0" | Input/Display
|Diameter of the mill (inside liners).
|-
|BellyLength
|style="background: #eaecf0" | Input/Display
|Length of the cylindrical section (belly) of the mill (inside liners).
|-
|TrunnionDiameter
|style="background: #eaecf0" | Input/Display
|Diameter of the trunnion (inside liners).
|-
|ConeAngle
|style="background: #eaecf0" | Input/Display
|Angular displacement of the cone surface from the vertical direction.
|-
|LifterHeight
|Input
|Height of the lifters above the liner surface
|-
|LifterAngle
|Input
|Angle between the face and the base of the lifter
|-
|MillRPM
|style="background: #eaecf0" | Input/Display
|Mill rotational speed
|-
|Jt
|style="background: #eaecf0" | Input/Display
|Volumetric fraction of the mill occupied by balls and coarse rock (including voids).
|-
|BallLoadVol
|style="background: #eaecf0" | Input/Display
|Volumetric fraction of the mill occupied by balls (including voids).
|-
|VoidFillFraction
|style="background: #eaecf0" | Input/Display
|Volumetric fraction of interstitial grinding media voidage occupied by slurry.
|-
|MeanMediaSize
|Input
|Characteristic media size, <math>\bar x</math>
|-
|SolidsSG
|style="background: #eaecf0" | Display
|Specific Gravity or density of solids.
|-
|LiquidsSG
|style="background: #eaecf0" | Display
|Specific Gravity or density of liquids.
|-
|BallSG
|style="background: #eaecf0" | Input/Display
|Specific Gravity or density of balls.
|-
|Cw
|style="background: #eaecf0" | Display
|Mass fraction of solids in discharge slurry.
|-
|DischargeType
|Grate/Overflow
|Discharge configuration, grate or overflow.
|-
|Gamma
|Input
|Value of the slip coefficient, <math>\gamma</math>
|-
|ThetaShoulder
|style="background: #eaecf0" | Display
|Angle of the shoulder of the outermost charge shell.
|-
|ThetaToe
|style="background: #eaecf0" | Display
|Angle of the toe of the outermost charge toe
|-
|NoLoadPower
|style="background: #eaecf0" | Display
|Power input to the motor when the mill is empty (no balls, rocks or slurry).
|-
|NetImpactPower
|style="background: #eaecf0" | Display
|Net power attributable to impact
|-
|NetAttAbrPower
|style="background: #eaecf0" | Display
|Net power attributable to attrition/abrasion
|-
|GrossPower
|style="background: #eaecf0" | Display
|Power input to the motor.
|}

== See also ==

* [[Tumbling Mill (Power, Morrell Continuum)|Morrell Continuum model]]
* [[Tumbling Mill (Power, Morrell Empirical)|Morrell Empirical model]]

== References ==

[[Category:Excel]]
[[Category:SysCAD]]