Reference ID: MET-CC5C | Process Engineering Reference Sheets Calculation Guide
Introduction & Context
In crystallisation processes, the rate at which a crystal grows is often governed by the
transport of solute to the crystal surface and the kinetics of incorporation at that
surface. When the mass transfer to the surface is fast compared with the surface
reaction, the growth is said to be surface integration controlled. The
calculation presented here provides a simple, empirical expression for the crystal
growth rate that can be used in process design, scale-up, and optimisation of
crystallisation units such as crystallisers, crystallisation columns, and
crystallisation reactors in the pharmaceutical, chemical, and materials
industries.
Methodology & Formulas
The calculation follows a standard empirical approach:
Supersaturation is defined as the difference between the bulk solute
concentration and the saturation concentration at the operating temperature:
\[
\Delta C = C - C_{\text{s}}
\]
A small positive value is enforced to avoid division by zero or negative
supersaturation.
Growth rate is expressed as a power-law function of supersaturation:
\[
G = k \, (\Delta C)^{g}
\]
where \(k\) is an empirical rate constant with units of
\(\text{g}\,\text{cm}^{-2}\,\text{h}^{-1}\,(\text{mol}\,\text{L}^{-1})^{-g}\) and
\(g\) is the growth order (dimensionless). This form captures the
non-linear dependence of the growth rate on supersaturation that is
characteristic of surface integration controlled growth.
Validity checks are performed to ensure that the input parameters
lie within empirically established bounds. If any parameter falls outside
its typical range, a warning is issued. The bounds are summarised in the
table below.
Parameter
Typical Range
k (rate constant)
\(0.1 \le k \le 10.0\) \(\text{g}\,\text{cm}^{-2}\,\text{h}^{-1}\,(\text{mol}\,\text{L}^{-1})^{-g}\)
g (growth order)
\(1.0 \le g \le 2.0\)
ΔC (supersaturation)
\(0.01 \le \Delta C \le 0.5\) \(\text{mol}\,\text{L}^{-1}\)
T (temperature)
\(0.0 \le T \le 80.0\) °C
P (pressure)
\(0.5 \le P \le 2.0\) bar
Answer:
In surface integration controlled growth, the rate at which atoms or molecules are incorporated into the crystal lattice is limited by the rate at which they can attach to the crystal surface. The growth rate, R, is expressed as:
\[ R = k_s \cdot (C - C_{\text{eq}}) \]
where \(k_s\) is the surface integration coefficient, \(C\) is the bulk concentration of the growth species, and \(C_{\text{eq}}\) is the equilibrium concentration. Because the process is controlled by surface kinetics, changes in surface conditions (e.g., roughness, defect density) directly influence \(k_s\) and thus the overall growth rate.
Answer:
Measure the steady-state growth rate, R, at several known supersaturations (C - Ceq).
Plot R versus (C - Ceq). The slope of the linear fit gives ks.
Verify linearity; deviations may indicate additional transport limitations.
Repeat at different temperatures to confirm Arrhenius behavior of ks.
Answer:
Surface integration coefficient (ks) – depends on surface structure and defect density.
Supersaturation (C - Ceq) – drives the driving force for attachment.
Temperature – affects both ks and the equilibrium concentration Ceq.
Surface morphology – roughness and step density can enhance or inhibit attachment.
Presence of impurities or additives that block active sites.
Answer:
Express ks as an Arrhenius function: \(k_s = k_0 \cdot \exp(-E_a / (R_g \cdot T))\), where \(E_a\) is the activation energy for surface integration, \(R_g\) is the gas constant, and \(T\) is absolute temperature.
Update Ceq with temperature using the appropriate solubility or vapor pressure data.
Recalculate R at each temperature using the modified ks and Ceq values.
Validate the model by comparing predicted growth rates with experimental measurements across the temperature range.
Worked Example – Surface-Integration Controlled Crystal Growth
A bench-scale cooling crystalliser is used to grow pharmaceutical-grade β-lactam crystals from an aqueous solution. The growth rate is known to be limited by surface integration. Determine the linear growth rate G under the following operating conditions.
Knowns
Temperature, T = 25.0 °C
Absolute pressure, P = 1.0 bar
Bulk solute concentration, C = 0.120 kg kg−1
Saturation concentration, Cs = 0.100 kg kg−1
Surface-integration rate constant, k = 0.500 m s−1 (kg kg−1)−1
Order of surface integration, g = 1.0 (dimensionless)
Step-by-Step Calculation
Convert temperature to kelvin for consistency:
\[ T_{\text{K}} = T + 273.15 = 25.0 + 273.15 = 298.15\ \text{K} \]
Compute the supersaturation driving force:
\[ \Delta C = C - C_{\text{s}} = 0.120 - 0.100 = 0.020\ \text{kg kg}^{−1} \]
Apply the surface-integration growth-rate model:
\[ G = k\ (\Delta C)^{g} \]
Substitute the numerical values:
\[ G = 0.500\ \text{m s}^{−1}\ (\text{kg kg}^{−1})^{−1} \times (0.020\ \text{kg kg}^{−1})^{1.0} \]
