Reference ID: MET-22A8 | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
The diffusion-controlled crystal growth rate calculation estimates how fast a solid phase (e.g., NaCl) precipitates from a supersaturated liquid when mass transfer to the crystal surface is the rate-limiting step.
This type of analysis is essential in process engineering for designing crystallizers, evaporators, and seawater desalination units, where controlling crystal size distribution, product purity, and equipment sizing depends on accurate growth-rate predictions.
Methodology & Formulas
The procedure follows a sequence of dimensionless groups that describe fluid flow and mass transfer, culminating in a mass-transfer coefficient and the crystal growth rate.
Temperature conversion – Convert the operating temperature from Celsius to Kelvin:
\[
T = T_{\mathrm{C}} + 273.15
\]
Viscosity conversion – Convert the dynamic viscosity from centipoise to pascal-seconds:
\[
\mu = \mu_{\mathrm{cP}} \times 10^{-3}
\]
Reynolds number (flow regime) – Ratio of inertial to viscous forces:
\[
\mathrm{Re} = \frac{\rho\,u\,D}{\mu}
\]
Schmidt number (momentum vs. mass diffusion) – Ratio of momentum diffusivity to mass diffusivity:
\[
\mathrm{Sc} = \frac{\mu}{\rho\,D_{AB}}
\]
Sherwood number (convective mass transfer) – Empirical correlation for turbulent pipe flow (generic form):
\[
\mathrm{Sh} = C_{1}\,\mathrm{Re}^{m}\,\mathrm{Sc}^{n}
\]
where \(C_{1}\), \(m\), and \(n\) are correlation constants.
Mass-transfer coefficient – Relates the Sherwood number to a dimensional coefficient:
\[
k_{d} = \frac{\mathrm{Sh}\;D_{AB}}{D}
\]
Crystal growth rate (diffusion-controlled) – Product of the mass-transfer coefficient and the supersaturation driving force:
\[
G = k_{d}\,\bigl(C_{\mathrm{bulk}} - C_{\mathrm{eq}}\bigr)
\]
Validity Checks & Regime Criteria
Condition
Criterion
Implication
Flow regime
\(\mathrm{Re} \;<\; \mathrm{Re}_{\text{crit}}\)
Laminar flow – the turbulent Sherwood correlation may over-predict mass transfer.
Schmidt number
\(\mathrm{Sc} \;>\; \mathrm{Sc}_{\text{min}}\)
Ensures that the correlation, derived for high-Sc fluids, remains applicable.
Supersaturation
\(C_{\mathrm{bulk}} \;>\; C_{\mathrm{eq}}\)
Necessary driving force for crystal growth; otherwise no precipitation occurs.
Sherwood number
\(\mathrm{Sh} \;>\; 0\)
Positive mass-transfer coefficient; a non-positive value indicates inconsistent inputs.
When all criteria are satisfied, the calculated growth rate \(G\) can be used to size crystallizer residence times, estimate product throughput, or evaluate the impact of operating conditions on crystal morphology.
The most common form of the diffusion-controlled growth rate is:
R = (D · (Cbulk – Ceq)) / δ
where R is the radial growth rate (m s⁻¹), D is the diffusion coefficient of the solute in the solvent (m² s⁻¹), Cbulk is the bulk concentration of the solute (mol m⁻³), Ceq is the equilibrium concentration at the crystal surface (mol m⁻³), and δ is the thickness of the diffusion boundary layer (m). This expression assumes steady-state diffusion and that surface kinetics are fast compared with diffusion.
To obtain D you can:
Search the literature for reported values under similar temperature and composition.
Estimate using the Stokes–Einstein relation: D = kB·T / (6π·η·r), where kB is Boltzmann’s constant, T is absolute temperature, η is solvent viscosity, and r is the hydrodynamic radius of the diffusing species.
Measure directly with tracer diffusion experiments (e.g., radioactive or isotopic labeling) and fit the concentration profile.
Adjust the value for temperature using an Arrhenius expression: D = D0·exp(–Ea/(Rg·T)).
Supersaturation, defined as (Cbulk – Ceq)/Ceq, directly influences the driving force for diffusion:
Higher supersaturation increases the concentration gradient, thereby raising R proportionally.
It also affects the thickness of the diffusion boundary layer; vigorous stirring or convection can reduce δ, further accelerating growth.
Excessive supersaturation may lead to secondary nucleation, complicating the growth regime.
Validation can be performed by:
Measuring crystal size over time using optical microscopy or laser diffraction and comparing the slope to the predicted R.
Monitoring the concentration of the solute in the bulk solution with in-situ spectroscopy to confirm the expected depletion rate.
Using a controlled diffusion cell (e.g., a Hele–Shaw cell) to isolate diffusion effects and directly observe the growth front.
Cross-checking with computational fluid dynamics (CFD) simulations that incorporate the same diffusion parameters.
Worked Example – Estimating Growth Rate of NaCl Crystals in a Draft-Tube Crystallizer
A 0.5 m ID draft-tube crystallizer is fed with a 25 °C brine that is 7.0 kg NaCl m-3 supersaturated. Laboratory data show the equilibrium solubility is 6.0 kg m-3. The vessel is agitated so that the average liquid velocity past the crystal faces is 0.5 m s-1. Estimate the diffusion-controlled linear growth rate of the crystals.
Knowns
Temperature: 25 °C (298.15 K)
Bulk concentration: Cbulk = 7.0 kg m-3
Equilibrium concentration: Ceq = 6.0 kg m-3
Diffusion coefficient of NaCl in water: DAB = 1.5 × 10-9 m2 s-1
Characteristic length (impeller region): L = 0.5 m
Velocity past crystal surface: u = 0.5 m s-1
Solution density: ρ = 1025 kg m-3
Solution viscosity: μ = 0.0011 Pa·s (1.1 cP)
Universal gas constant: R = 8.314 J mol-1 K-1
Step-by-step calculation
Compute the Reynolds number for flow past the crystal:
\[
Re = \frac{\rho u L}{\mu} = \frac{1025 \times 0.5 \times 0.5}{0.0011} = 9318
\]
Estimate the Sherwood number using the Ranz–Marshall correlation for forced convection:
\[
Sh = 2 + 0.6\,Re^{0.5}\,Sc^{0.33} = 2 + 0.6 \times (9318)^{0.5} \times (715)^{0.33} = 308
\]
