LABORATORY 45 · CHEMICAL & PROCESS ENGINEERING

Continuous Crystallization Population Balance Modeling & MSMPR Crystal Size Distribution

Operate an instrumented continuous mixed-suspension mixed-product-removal (MSMPR) cooling crystallizer for potassium sulfate (K_2SO_4) at steady state, analyze crystal size distribution (CSD) across sieve aperture sizes L from 50.0 to 300.0 μm using laser diffraction, evaluate the population density decay slope S, and determine crystal growth rate G and nucleation rate B_0. Explore interactive transport phenomena, unit operations simulation, and chemical engineering telemetry.

TRANSPORT PHENOMENA · UNIT OPERATIONS · PROCESS CONTROL · REACTION ENGINEERING

Your virtual chemical and process engineering laboratory.

CRYSTAL POPULATION DENSITY LN[N(L)]—

PROCESS & PILOT PLANT TELEMETRY

Ready to sample with Crystal Size L = 150.0 μm.

Crystal Characteristic Sieve Size L150.0 μmSet Crystal Size L (μm)

Process Fluid: Standard industrial test fluid at 20.0 °C

Adjust Crystal Size L, then execute chemical process trial.

Chemical and process engineering telemetry is evaluated rigorously using 64-bit IEEE 754 precision. Residuals reflect ordinary least-squares regression across successive observations. Continuous data structures support high-fidelity transport phenomena and unit operations modeling.

Process engineering assumptions and boundary condition limits

The transport and thermodynamic models apply within the specified calibrated operating window (50.0 to 300.0 μm). Evaluated using a 4.0 L draft-tube baffled (DTB) cooling MSMPR crystallizer operating at steady state (τ = 40.0 min, agitation 450 RPM, jacket temperature 20.0 °C, slurry temperature 25.00 ± 0.05 °C). K_2SO_4 aqueous system analyzed via Malvern Mastersizer 3000 laser diffraction and calibrated analytical sieves.

Process apparatus & industrial sensors.

4.0 L Glass Draft-Tube Baffled (DTB) MSMPR Crystallizer

Vessel with internal draft tube, 4 wall baffles, and bottom cooling jacket providing uniform suspension.

Low-Shear Marine Propeller Drive with Tachometer

Provides controlled axial up-pumping at 450 ± 2 RPM to keep crystals suspended without attrition.

Recirculating Thermostatic Chiller Bath

Controls cooling jacket at 20.00 ± 0.02 °C to maintain constant supersaturation driving force ΔC.

Dual Peristaltic Slurry Feed and Withdrawal Pumps

Delivers saturated K_2SO_4 feed and withdraws mixed slurry at matched 100.0 mL/min (τ = 40.0 min).

Malvern Mastersizer 3000 Laser Diffraction Particle Size Analyzer

Measures volumetric and population crystal size distributions across 0.1 to 1000 μm in real time.

Precision Analytical Sieve Shaker with Sonic Sifter

Separates washed, ethanol-rinsed crystal batches across calibrated 50, 100, 150, 200, 250, 300 μm meshes.

FOLLOW THE METHOD

A reason behind every step.

Operate an instrumented continuous mixed-suspension mixed-product-removal (MSMPR) cooling crystallizer for potassium sulfate (K_2SO_4) at steady state, analyze crystal size distribution (CSD) across sieve aperture sizes L from 50.0 to 300.0 μm using laser diffraction, evaluate the population density decay slope S, and determine crystal growth rate G and nucleation rate B_0. Standardized parameter settings: 50.0, 100.0, 150.0, 200.0, 250.0, 300.0 μm.

Feedstock Preparation & Crystallizer Priming

Dissolve reagent-grade K_2SO_4 in DI water at 40.0 °C to make saturated feed (14.0 g/100 g H_2O); fill crystallizer with 4.0 L solution and stabilize at 25.0 °C.

Continuous Flow Initiation (τ = 40.0 min)

Set feed pump to deliver 100.0 mL/min (40 °C feed) and withdrawal pump to remove 100.0 mL/min slurry (residence time τ = 40.0 min).

Steady-State Magma Equilibration

Operate continuously for 12 residence times (8.0 hours) to allow crystal nucleation, growth, and slurry magma density to reach true dynamic steady state.

Isokinetic Slurry Sampling & Filtration

Withdraw 250 mL representative magma samples from the draft-tube exit using an isokinetic sampling probe; vacuum filter immediately and wash with cold ethanol.

Laser Diffraction & Sieve Size Analysis

Analyze wet crystal dispersion via laser diffraction; dry crystals at 60 °C and run sonic sieving across 50, 100, 150, 200, 250, and 300 μm sieves.

Population Balance Density Regression

Calculate population densities n(L) = ΔM / (ρ_c k_v L^3 ΔL V_sample); plot ln[n(L)] versus L to determine decay slope S and growth rate G.

YOUR NOTEBOOK · 0 OBSERVATIONS

Turn measurements into chemical process engineering insight.

See a complete example report ↗

Points are your recorded observations. The solid regression line validates MSMPR Continuous Crystallization mathematical scaling.

LINEAR REGRESSION FIT · y = mx + b

ln[n(L)] = -S · L + c_0 = -0.0125 · L + 15.2000

#	Crystal Characteristic Sieve Size L (μm)	Crystal Population Density ln[n(L)] (ln(no./(μm·mL)))	Crystal characteristic size L (μm)	Natural logarithm of population density ln[n(L)]	Remove

No readings yet. Execute a process trial and record it.

Calculate it yourself.

Your estimate of MSMPR Population Decay Slope S (μm^-1)

CHECK YOUR UNDERSTANDING

Explain what the process engineering metrics mean.

In an ideal MSMPR continuous crystallizer obeying McCabe's ΔL law, the population density n(L) follows:

On a semi-log plot of ln[n(L)] versus crystal size L, the slope S is equal to:

If MSMPR residence time is τ = 40.0 min and the population decay slope is S = 0.0125 μm^-1, the crystal growth rate G is:

The zero-size intercept n_0 on an MSMPR population plot represents:

Secondary nucleation in a stirred MSMPR crystallizer is primarily generated by:

THEORY AND UNIT OPERATIONS CONTEXT

Process engineering background & physical principles.

Industrial crystallization from solution is a primary separation and purification unit operation in chemical, fertilizer, pharmaceutical, and specialty materials manufacturing. In continuous crystallization, product crystal size distribution (CSD) directly determines downstream filtration cake porosity, washability, drying kinetics, pneumatic conveying flowability, and bulk dissolution rates. The benchmark theoretical framework for continuous industrial crystallization is the Mixed-Suspension Mixed-Product-Removal (MSMPR) crystallizer formalized by Alan Randolph and Maurice Larson in 1971. In an ideal MSMPR crystallizer, steady-state continuous feed enters an intensely agitated vessel (mean residence time τ = V / v_0 = 40.0 min), supersaturation is generated continuously (via cooling or solvent evaporation), crystals nucleate and grow throughout the suspension, and the exit slurry has an identical composition and particle size distribution to the magma inside the vessel. Applying the macroscopic one-dimensional Population Balance Equation (PBE) under steady-state conditions with size-independent linear crystal growth (McCabe's ΔL law, dG/dL = 0) and negligible crystal breakage or agglomeration yields: d(n · G)/dL + n / τ = 0, where n(L) is the crystal population density (number of crystals per unit size per unit slurry volume, no./(μm·mL)), G is the linear crystal growth rate (dL/dt, μm/min), and τ is the crystallizer residence time (min). Integrating this first-order ODE subject to boundary condition n(0) = n_0 = B_0 / G (where n_0 is nuclei population density and B_0 is primary nucleation rate, no./(min·mL)) gives the exponential population density distribution: n(L) = n_0 · exp[-L / (G · τ)]. Taking natural logarithms linearizes the MSMPR population model: ln[n(L)] = -S · L + c_0, where the population decay slope is S = 1 / (G · τ) = 0.0125 μm^-1 and the intercept is c_0 = ln(n_0) = 15.2000. With known residence time τ = 40.0 min, the linear crystal growth rate is evaluated directly as G = 1 / (S · τ) = 2.0000 μm/min. In this laboratory experiment, a 4.0-liter continuous cooling MSMPR crystallizer precipitating potassium sulfate (K_2SO_4) from aqueous solution at 25.0 °C is operated at steady state to sample crystal slurries across size fractions from 50.0 to 300.0 μm, verify the log-linear population decay, and extract G = 2.0000 μm/min and B_0 = 7.97 × 10^6 no./(min·mL).

ln[n(L)] = -S · L + c_0 = -0.0125 · L + 15.2000

Evaluated using a 4.0 L draft-tube baffled (DTB) cooling MSMPR crystallizer operating at steady state (τ = 40.0 min, agitation 450 RPM, jacket temperature 20.0 °C, slurry temperature 25.00 ± 0.05 °C). K_2SO_4 aqueous system analyzed via Malvern Mastersizer 3000 laser diffraction and calibrated analytical sieves.