Chapter 10

Solids Operations

Every unit op so far has treated a stream as a single homogeneous fluid. The moment a solid forms — crystals in a leach liquor, catalyst fines, a precipitated hydroxide cake — a whole second physics kicks in: particles have a size, and how big they are decides how fast you can filter them, how wet the resulting cake stays, and how much heat it takes to dry it. This chapter follows one crystal from the mother liquor to a dry powder.

10.1 Crystallization: nucleation, growth, and the MSMPR

A crystal doesn't appear at a fixed size — a population of them nucleates and grows together. The workhorse model is the MSMPR (mixed-suspension, mixed-product-removal) crystallizer: nucleation rate

B

and linear growth rate

G

both driven by the same supersaturation

\Delta C = C - C_{sat}

, each with its own empirical order:

B = k_b\,\Delta C^{b},\qquad G = k_g\,\Delta C^{g}

$B$: Nucleation rate — number of new crystals born per unit volume per unit time (#/m³·s).
$G$: Linear growth rate — how fast a crystal's characteristic length grows (m/s).
$k_b,\; k_g$: Nucleation and growth rate constants — fitted per system, not universal (from lab crystallization data).
$b,\; g$: Empirical orders of the supersaturation dependence — typically b > g, which is why raising supersaturation nucleates faster than it grows (more, smaller crystals).
$\Delta C$: Supersaturation — the actual solute concentration C above the saturation concentration C_{sat} at the operating temperature; the driving force for both B and G.

Rather than track every crystal size individually, the MSMPR model works in moments of the size distribution —

m_0

(crystal count),

m_1

(total length),

m_2

(area-related),

m_3

(volume, i.e. mass) — each moment built from the one below it over the residence time

\tau

m_0 = B\tau,\quad m_1 = m_0 G \tau,\quad m_2 = 2 m_1 G \tau,\quad m_3 = 3 m_2 G \tau,\qquad \bar L = m_1/m_0 = G\tau

$m_0,\, m_1,\, m_2,\, m_3$: The zeroth through third moments of the crystal-size distribution — number, total length, area-related, and volume/mass-related, each built from the one below it via the growth rate G over the residence time τ.
$\tau$: Residence time — how long a crystal spends in the well-mixed crystallizer volume before leaving with the product stream (s).
$\bar L$: Mean crystal size — the population-average characteristic length, m_1/m_0, which collapses to Gτ for the MSMPR's simple moment cascade.

Worked example — mean crystal size

A crystallizer growth rate

G = 2\times10^{-8}

m/s and residence time

\tau = 3600

s give mean crystal size:

\bar L = G\tau = (2\times10^{-8}\ \text{m/s})(3600\ \text{s}) = 7.2\times10^{-5}\ \text{m} = 72\ \mu\text{m}

A run twice as long (or grown at half the rate) doubles it — the direct lever a plant pulls when downstream filtration needs a coarser, faster-draining cake (§10.6 warm-up exercise below reproduces this exact number).

The mean crystal size

\bar L

falls right out as

G\tau

— a crystallizer run longer (or grown slower) makes bigger crystals, which is exactly the lever a plant pulls when downstream filtration needs a coarser, faster-draining cake. The solid production rate and the crystallization duty follow from the third moment and the (usually exothermic) heat of crystallization:

\dot m_{solid} = \rho_c\,k_v\,m_3,\qquad Q = -\dot n_{solid}\,\Delta H_{cryst}

$\dot m_{solid}$: Solid mass production rate leaving the crystallizer (kg/s).
$\rho_c$: Crystal (solid-phase) density — mass per unit volume of the pure crystal (kg/m³).
$k_v$: Volumetric shape factor — how efficiently the crystal shape fills its bounding cube (k_v = 1 for a perfect cube; real crystals pack less efficiently, so k_v < 1).
$\dot n_{solid}$: Molar solid production rate (mol/s) — the same physical flow as ṁ_solid, in moles rather than mass.
$\Delta H_{cryst}$: Molar heat of crystallization (J/mol) — usually exothermic (negative), so Q comes out as heat that must be removed to hold the crystallizer at temperature.

k_v

is a shape factor (a cube has

k_v=1

; real crystals are less efficient packers). MaximaLabs' crystallizer unit op solves exactly this moment model — set the saturation concentration, feed concentration, residence time, and heat of crystallization, and it returns the solid stream (with its mean size) and the mother liquor. Beyond the moment model, a distributed population-balance mode (a full size-grid solve with an optional agglomeration kernel) is also available for cases where the moment model's implicit single-mode distribution isn't enough — see the "further reading" note at the end of this chapter.

10.2 Solid-liquid separation: cake filtration

Crystals leave the crystallizer suspended in liquor — a filter (or centrifuge) splits that suspension into a wet solid cake and a clear filtrate. Two things matter: how much solid actually reports to the cake (the recovery) and how much liquid stays trapped in it (the cake moisture):

\text{cake} = R\,\dot n_s + \dot n_{liq,ret},\qquad m = \dot n_{liq,ret}/\text{cake}

$R$: Specified solids recovery — the fraction of the feed's solid that reports to the cake (0–1).
$m$: Specified cake moisture fraction — the fraction of the cake's total mass that is retained liquid.
$\dot n_s$: Molar (or mass) flow of solid arriving at the filter in the feed suspension.
$\dot n_{liq,ret}$: Liquid flow retained (trapped) in the cake — everything else in the feed liquid reports to the filtrate.

where

R

is the specified solids recovery and

m

the specified cake moisture fraction — the rest of the liquid, plus any unrecovered solid, reports to the filtrate. Sizing the actual filter area for a given cake filtration time

t_c

and pressure drop

\Delta P

uses the classic Ruth cake-filtration equation, built on the same specific cake resistance

\alpha

and slurry concentration

c

a lab filter-leaf test measures:

A = Q_f\sqrt{\dfrac{\mu\,\alpha\,c\,t_c}{2\,\Delta P}}\qquad(\text{Ruth cake filtration})

$A$: Required filter area (m²) — the unknown this equation is solved for.
$Q_f$: Volumetric feed (slurry) flow rate to the filter (m³/s).
$\mu$: Filtrate (liquid) viscosity (Pa·s).
$\alpha$: Specific cake resistance — how much a unit mass of deposited cake resists flow (m/kg); a lab filter-leaf test measures this per system, it isn't a universal constant.
$c$: Slurry solids concentration — mass of solid per unit volume of filtrate (kg/m³).
$t_c$: Cake filtration time — how long the filter runs per cycle/batch (s).
$\Delta P$: Pressure drop driving flow through the cake and filter medium (Pa).

A centrifuge separates the same suspension by replacing gravity with centrifugal acceleration —

\Sigma

theory converts a centrifuge's geometry and rotation speed into an equivalent gravity-settler area, letting you compare machines on the same basis:

v_g = \frac{G\,g\,(\rho_s-\rho_l)\,d^2}{18\,\mu},\qquad \Sigma = \frac{Q_f}{2\,v_g}

$v_g$: Gravity-settling (terminal) velocity of a particle of diameter d under Stokes' law (m/s).
$G$: g-force multiple — the centrifuge's rotational acceleration divided by gravitational acceleration (dimensionless), from its bowl radius and rotation speed.
$g$: Gravitational acceleration, 9.81 m/s².
$\rho_s,\; \rho_l$: Solid particle density and liquid (continuous-phase) density (kg/m³) — their difference is the buoyancy-corrected driving force.
$d$: Particle diameter (m).
$\Sigma$: Equivalent settling area — the gravity-clarifier area that would give the same separation performance as the centrifuge, letting different machines be compared on one basis (m²).

10.3 Drying: the last of the liquid

The filter cake still carries the specified moisture — a dryer evaporates the rest down to a target outlet moisture

m_{out}

. The material balance on the liquid picks off exactly the evaporated amount

E

, and the duty is that evaporation rate times the latent heat of vaporization at the drying temperature:

E = L_{in} - S\,\dfrac{m_{out}}{1-m_{out}},\qquad Q = E\,\big(h_v(T) - h_l(T)\big)

$E$: Evaporated liquid flow — the amount removed from the cake to hit the target outlet moisture (mol/s).
$L_{in}$: Incoming liquid flow carried in the feed cake.
$S$: Dry-solid flow — constant through the dryer (solids don't evaporate).
$m_{out}$: Target outlet moisture fraction (liquid mass / total mass) — the spec the dryer solves E for.
$Q$: Drying duty — the heat input required (W).
$h_v(T),\; h_l(T)$: Vapor- and liquid-phase molar enthalpy of the liquid component at the drying temperature T, from the active thermo package — their difference is the latent heat of vaporization.

S

is the (constant, non-evaporating) dry-solid flow and

L_{in}

the incoming liquid — so the dryer solves for how much water has to leave to hit the target moisture, and prices that in real latent-heat duty from the active thermo package, not a rule-of-thumb energy number.

10.4 Sizing and classification: mills, screens, cyclones

Not every solids problem is about removing liquid — sometimes the goal is changing particle size, or splitting a distribution by size. A mill (comminution) reduces particle size and its power draw follows Bond's law, empirically relating the specific energy to the 80%-passing sizes before (

F_{80}

) and after (

P_{80}

) grinding:

W = 10\,W_i\left(\frac{1}{\sqrt{P_{80}}} - \frac{1}{\sqrt{F_{80}}}\right)\qquad(\text{Bond work index})

$W$: Specific grinding energy required (kWh/t).
$W_i$: Bond work index — an empirical, material-specific measure of grindability (kWh/t), measured by a standard lab grinding test; higher means harder to grind.
$F_{80},\; P_{80}$: 80%-passing particle size of the feed and product — the size below which 80% of the material's mass lies, before and after grinding (μm), Bond's chosen way to characterize a whole size distribution with one number.

A screen splits a particle-size distribution at a cut size using a log-normal partition (sharper for a well-designed screen, fuzzier for a real one via

\sigma_g

), and a cyclone separates fine solids from a gas by centrifugal action, characterized by its cut diameter

d_{50}

(the Lapple correlation) and a partition efficiency that falls off around it:

F_{under} = \Phi\!\left(\frac{\ln(c/d_{50})}{\ln\sigma_g}\right)\qquad(\text{screen partition})

d_{50} = \sqrt{\dfrac{9\mu W}{2\pi N_e v\,(\rho_p-\rho_g)}},\qquad \eta = \dfrac{1}{1+(d_{50}/d_{eff})^2}\qquad(\text{cyclone, Lapple})

$F_{under}$: Partition fraction reporting to the undersize (screened-through) stream, for a particle of size c.
$\Phi$: Standard normal cumulative distribution function — the log-normal partition curve's S-shape comes from evaluating it in log-size space.
$c$: Particle size being partitioned (the screen equation's own variable, not the slurry concentration used earlier in §10.2).
$d_{50}$: Cut diameter — the particle size at which the screen (or cyclone) splits 50/50 between the two outlet streams.
$\sigma_g$: Geometric standard deviation of the partition curve — how sharp (near 1) or fuzzy (larger) the cut is; a real screen is never a perfect knife-edge.
$\mu$: Gas viscosity, for the cyclone (Pa·s).
$W$: Cyclone inlet width (m) — reuses the same symbol as Bond's specific energy above but in a different unit-op context, per the Lapple correlation's own notation.
$N_e$: Effective number of turns the gas makes inside the cyclone body (dimensionless, geometry-dependent).
$v$: Cyclone inlet gas velocity (m/s).
$\rho_p,\; \rho_g$: Particle and gas density (kg/m³).
$\eta$: Collection (partition) efficiency for a particle of the effective diameter d_{eff}, falling off around d_{50}.

All four — mill, screen, cyclone, plus filter/centrifuge above — carry a full solid-phase payload (particle size, moisture, composition) on the stream itself, so a mill feeding a screen feeding a cyclone is a genuine size-tracking train, not three independent black boxes.

10.5 Try it

Reproduce it in your browser

1Open the solids train example and Run it: FEED → CR (MSMPR crystallizer) → FIL (cake filter) → DRY (dryer).
2Click the CR node and open the Theory tab — the exact nucleation/growth/moment equations of §10.1, with the solved mean crystal size and duty for this stream.
3Raise the crystallizer's residence time and re-run — the mean crystal size $\bar L = G\tau$ grows, exactly as §10.1 predicts. Then check the FIL node: a coarser crystal typically filters faster and drier in a real plant, even though this screening-level filter model takes recovery/moisture as direct specs rather than deriving them from crystal size.
4Tighten the dryer's outlet moisture spec and re-run — watch the dryer duty in the stream table rise, since more water now has to evaporate per §10.3.

The moment-based crystallizer above is a screening model — MaximaLabs also has a distributed population-balance mode with an optional agglomeration kernel for cases where the crystal-size distribution itself (not just its mean and count) is the answer you need, e.g. sizing a filter to a real, possibly bimodal, distribution rather than an assumed mean.

10.6 Exercises

Practice

Work each problem yourself first, then reveal the solution to check it. Where a problem says so, reproduce it live in MaximaLabs — the solver is the answer key.

1
warm-up
An MSMPR crystallizer has growth rate $G = 2\times10^{-8}$ m/s and a residence time $\tau = 3600$ s. Estimate the mean crystal size $\bar L$ .
2
core
Two crystallizers hit the same mean crystal size $\bar L=G\tau$ — one by running twice as long ( $\tau$ ) at half the growth rate $G$ , the other by running half as long at double the growth rate. Are their crystal-count populations $m_0 = B\tau$ also the same? What does that imply about their filtration behavior even at equal mean size?
3
challenge
In the solids train, the dryer's outlet-moisture spec is tightened from 10% to 1%. Using §10.3's equation, explain qualitatively why the marginal evaporation duty needed per percentage point gets larger as the target moisture approaches zero, not constant.

Open the solids train example →← Chapter 9 Chapter 11 →The Handbook