Agglomeration Diagnosis in Crystallizers

Reference ID: MET-1EAE | Process Engineering Reference Sheets Calculation Guide

Agglomeration Diagnosis in Crystallizers

Introduction & Context

Agglomeration in crystallizers refers to the unwanted clustering of crystals into larger particles due to collision and adhesion, rather than controlled growth. This phenomenon is critical in process engineering, particularly in industries such as pharmaceuticals, food processing, and chemical manufacturing, where precise control over Crystal Size Distribution (CSD) is essential for product quality, downstream processing, and efficiency.

This diagnostic tool evaluates whether observed crystal sizes in a crystallizer are primarily due to growth-driven mechanisms (desirable) or agglomeration (often undesirable). It integrates fluid dynamics (shear stress, residence time), crystallization kinetics (growth rate, supersaturation), and process conditions (agitator speed, viscosity) to identify potential agglomeration risks. Typical applications include:

Batch/Continuous Crystallizers: Diagnosing deviations from target CSD.
Scale-Up/Scale-Down: Predicting agglomeration behavior across different vessel sizes.
Process Optimization: Balancing supersaturation, shear, and residence time to minimize agglomeration.
Troubleshooting: Identifying root causes of unexpected large particles (e.g., sticky collisions vs. growth).

Methodology & Formulas

1. Residence Time ($ t_{\text{res}} $)

The residence time represents the average time a crystal spends in the crystallizer, influencing its growth. For a continuously stirred tank crystallizer (CSTC), it is calculated as the ratio of the vessel volume ($ V $) to the volumetric feed flow rate ($ Q $):

\[ t_{\text{res}} = \frac{V}{Q} \]

where:

$V = \pi R^2 H$ is the crystallizer volume (cylindrical geometry),
$$ R $$ is the crystallizer radius,
$$ H $$ is the crystallizer height,
$$ Q $$ is the volumetric feed flow rate.

2. Shear Rate ($ \dot{\gamma} $)

The shear rate characterizes the velocity gradient in the shear zone near the agitator. For a linear shear profile (valid under laminar conditions), it is approximated as:

\[ \dot{\gamma} = \frac{U_{\text{agit}}}{h} \]

where:

$U_{\text{agit}}$ is the agitator tip speed,
$$ h $$ is the shear zone thickness.

3. Shear Stress ($ \tau $)

Shear stress quantifies the force per unit area acting on crystals due to fluid motion. It is the product of the fluid viscosity ($ \mu $) and the shear rate:

\[ \tau = \mu \dot{\gamma} \]

Low shear stress may fail to break agglomerates, while excessive shear can cause attrition or secondary nucleation.

4. Growth-Only Crystal Size ($ d_{\text{growth}} $)

The theoretical crystal size due to growth alone (no agglomeration) is estimated by multiplying the crystal growth rate ($ k_g $) by the residence time:

\[ d_{\text{growth}} = k_g \cdot t_{\text{res}} \]

where $ k_g $ depends on supersaturation, temperature, and crystal properties.

5. Reynolds Number for Shear Zone ($ \text{Re}_{\text{shear}} $)

The Reynolds number assesses the flow regime in the shear zone (laminar vs. turbulent). For the shear layer:

\[ \text{Re}_{\text{shear}} = \frac{\rho U_{\text{agit}} h}{\mu} \]

where:

$\rho$ is the fluid density,
$\mu$ is the fluid dynamic viscosity.

Flow Regime	Reynolds Number ($ \text{Re}_{\text{shear}} $)	Implications
Laminar	< 2000	Linear shear profile assumption holds. Agglomeration risk depends on $ \tau $ and supersaturation.
Transitional	2000–4000	Shear profile becomes nonlinear; empirical corrections may be needed.
Turbulent	> 4000	High shear fluctuations; agglomeration less likely but attrition risk increases.

6. Agglomeration Diagnosis Criterion

Agglomeration is likely if the actual measured crystal size ($ d_{\text{actual}} $) significantly exceeds the growth-only size ($ d_{\text{growth}} $). A threshold of 20% deviation is commonly used:

\[ \text{Agglomeration Likely if: } d_{\text{actual}} > 1.2 \cdot d_{\text{growth}} \]

Condition	Diagnosis	Root Causes	Mitigation Strategies
$d_{\text{actual}} \leq 1.2 \cdot d_{\text{growth}}$	No significant agglomeration	Balanced growth and shear	Maintain current conditions
$d_{\text{actual}} > 1.2 \cdot d_{\text{growth}}$	Agglomeration likely	High supersaturation ($ S > 1.2 $) → sticky collisions Low shear stress ($ \tau < 1 \, \text{Pa} $) → weak agglomerate breakup Long residence time → prolonged collision opportunities	Reduce supersaturation (adjust temperature/feed concentration) Increase shear (higher $ U_{\text{agit}} $ or lower $ h $) Add anti-agglomeration agents (e.g., surfactants) Shorten residence time (increase $ Q $)

7. Supersaturation Ratio ($ S $)

The supersaturation ratio ($ S $) is the driving force for crystallization, defined as:

\[ S = \frac{c}{c^*} \]

where:

$$ c $$ is the solute concentration,
$$ c^* $$ is the saturation concentration.

Supersaturation Ratio ($ S $)	Regime	Agglomeration Risk
$$ S < 1 $$	Undersaturated	No growth or agglomeration (crystals may dissolve)
$$ 1 < S < 1.1 $$	Low Supersaturation	Minimal risk; growth-dominated
$1.1 \leq S \leq 1.3$	Moderate Supersaturation	Balanced growth; agglomeration possible if shear is low
$$ S > 1.3 $$	High Supersaturation	High agglomeration risk; primary/secondary nucleation may dominate

Agglomeration often manifests through a combination of observable and measurable signs. Typical indicators include:

Sudden increase in pressure drop across the crystallizer or downstream filters.
Decrease in crystal size distribution uniformity, with a shift toward larger, irregular particles.
Reduced slurry flowability, leading to higher apparent viscosity.
Frequent fouling or plugging of heat transfer surfaces and internal baffles.
Abnormal torque readings on agitators or pumps, indicating higher resistance.

Distinguishing the two phenomena requires a systematic approach:

Observe the duration: bridging is usually transient and resolves when flow conditions change, whereas agglomeration persists.
Inspect the particle morphology using microscopy; agglomerates show fused crystal surfaces, while bridges consist of intact crystals merely in contact.
Monitor process variables (e.g., supersaturation, temperature) – a sudden spike often triggers bridging, while gradual trends favor agglomeration.
Perform a quick pressure-drop test: a reversible drop suggests bridging; a sustained increase points to agglomeration.

Real-time detection can be achieved with a combination of the following instruments:

In-line laser diffraction – provides continuous particle size distribution data.
Acoustic emission sensors – capture changes in slurry sound signatures associated with agglomerate formation.
Pressure transducers placed before and after the crystallizer – monitor pressure-drop trends.
Rheometers or torque meters integrated on the agitator shaft – detect viscosity spikes.
Vision-based cameras with image analysis – identify morphological changes on transparent sections.

Mitigation strategies depend on the root cause but generally include:

Adjust agitation speed or impeller type to increase shear and break up agglomerates.
Modify supersaturation levels by fine-tuning feed concentration or temperature to reduce crystal growth rates.
Introduce a controlled seed addition to promote uniform nucleation and limit secondary agglomeration.
Implement periodic back-flushing or pulse-jet cleaning of filters and heat-transfer surfaces.
Consider adding a dispersant or anti-agglomerant additive compatible with the process chemistry.

Worked Example – Detecting Agglomeration Risk in a Draft-Tube Crystallizer

A specialty-chemical plant runs a 150 mm ID draft-tube crystallizer that produces 400 µm sodium-chlorate crystals. Recent off-spec product (lumps > 1 mm) suggests unwanted agglomeration. The shift engineer wants a quick diagnostic: will the prevailing shear inside the vessel break the 400 µm primary particles apart, or do the conditions favour growth into agglomerates?

Knowns

Impeller tip speed $ U_\text{agit} $ = 0.5 m s⁻¹
Estimated shear-gap thickness $ h_\text{shear} $ = 10 mm = 0.01 m
Residence time $ t_\text{res} $ = 42.4 s
Supersaturation ratio $ S $ = 1.2
Growth rate constant $ k_\text{growth} $ = 5×10⁻⁶ m s⁻¹
Primary crystal size $ d_\text{actual} $ = 400 µm = 0.0004 m
Liquid density $ \rho_\text{cryst} $ = 998 kg m⁻³
Liquid viscosity $ \mu_\text{cryst} $ = 0.001 Pa·s

Step-by-step calculation

Compute the average shear rate in the impeller swept region: \[ \dot{\gamma} = \frac{U_\text{agit}}{h_\text{shear}} = \frac{0.5}{0.01} = 50\ \text{s}^{-1} \]
Estimate the shear stress imparted on a particle: \[ \tau_\text{shear} = \mu_\text{cryst}\,\dot{\gamma} = 0.001 \times 50 = 0.050\ \text{Pa} \]
Calculate the growth increment during one pass through the vessel: \[ \Delta d_\text{growth} = k_\text{growth}\,(S-1)\,t_\text{res} = 5\times10^{-6}\,(0.2)\,42.4 = 4.24\times10^{-5}\ \text{m} \approx 42\ \mu\text{m} \]
Compare the disruptive shear stress with the typical cohesive strength of ionic crystals (≈ 0.1–1 MPa). Here $ \tau_\text{shear} $ (0.05 Pa) is four to five orders of magnitude below the strength, so shear-induced breakage is negligible.
Because the growth increment (42 µm) is ~10% of the primary size and the shear is too weak to fragment the particles, the diagnostic flag is "high agglomeration risk".

Final Answer

Under the current operating conditions, the crystallizer delivers only 0.050 Pa of shear stress—far below the threshold needed to break 400 µm sodium-chlorate crystals—while the solute layer grows each crystal by 42 µm every 42 s. Consequently, particles remain in contact long enough to cement together, confirming that the observed oversize lumps are due to agglomeration, not inadequate nucleation or true growth rate.

"Un projet n'est jamais trop grand s'il est bien conçu." — André Citroën

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle

Agglomeration Diagnosis in Crystallizers

Agglomeration Diagnosis in Crystallizers

Introduction & Context

Methodology & Formulas

1. Residence Time (\( t_{\text{res}} \))

2. Shear Rate (\( \dot{\gamma} \))

3. Shear Stress (\( \tau \))

4. Growth-Only Crystal Size (\( d_{\text{growth}} \))

5. Reynolds Number for Shear Zone (\( \text{Re}_{\text{shear}} \))

6. Agglomeration Diagnosis Criterion

7. Supersaturation Ratio (\( S \))

Worked Example – Detecting Agglomeration Risk in a Draft-Tube Crystallizer

Supersaturation Ratio (\( S \))	Regime	Agglomeration Risk
$\( S < 1 \)$	Undersaturated	No growth or agglomeration (crystals may dissolve)
$\( 1 < S < 1.1 \)$	Low Supersaturation	Minimal risk; growth-dominated
$1.1 \leq S \leq 1.3$	Moderate Supersaturation	Balanced growth; agglomeration possible if shear is low
$\( S > 1.3 \)$	High Supersaturation	High agglomeration risk; primary/secondary nucleation may dominate