Introduction & Context

The Overall Crystal Growth Rate calculation is a fundamental process engineering tool used to predict the rate at which a solid phase grows from a supersaturated liquid solution. In industrial crystallization, this model is critical for designing crystallizers, determining continuous crystallizer residence time requirements, and controlling crystal size distribution (CSD). By employing a combined resistance model, engineers can account for both the transport of solute molecules through the liquid boundary layer (diffusion) and the subsequent incorporation of those molecules into the crystal lattice (surface integration). This calculation is typically applied in the pharmaceutical, food, and specialty chemical industries to ensure product quality and process efficiency.

Methodology & Formulas

The growth process is modeled as a series of resistances, with the total driving force defined as the difference between the bulk concentration and the saturation concentration. The presence of impurities alters these resistances, a phenomenon described in detail in the impurity effect on growth rate, which can significantly impact the overall crystal growth rate.

\[ \Delta C = C - C_{\text{sat}} \]

The mass transfer coefficient k_d is determined using the Ranz‑Marshall correlation, which accounts for the hydrodynamic conditions surrounding the crystal and is a key component of the diffusion‑controlled crystal growth rate calculation.

\[ Re = \frac{d_{p} \cdot v}{\nu} \] \[ Sc = \frac{\nu}{D_{AB}} \] \[ Sh = 2 + 0.6 \cdot Re^{1/2} \cdot Sc^{1/3} \] \[ k_{d} = \frac{Sh \cdot D_{AB}}{d_{p}} \]

The overall growth rate constant K combines the mass transfer coefficient k_d and the surface integration rate constant k_r, as detailed in the surface integration controlled crystal growth rate calculation.

\[ \frac{1}{K} = \frac{1}{k_{d}} + \frac{1}{k_{r}} \qquad\Longrightarrow\qquad K = \frac{k_{d} \cdot k_{r}}{k_{d} + k_{r}} \]

The mass flux R_G and the resulting linear growth rate G are calculated as follows:

\[ R_{G} = K \cdot \Delta C \] \[ G = \frac{R_{G}}{\rho_{c}} \]

Regime	Condition	Physical Significance
Diffusion Controlled	\( k_{d} \ll k_{r} \)	Mass transfer through the boundary layer is the bottleneck.
Surface Integration Controlled	\( k_{r} \ll k_{d} \)	Lattice incorporation kinetics limit the growth rate.
Ranz-Marshall Validity	\( 0.1 < Re < 1000 \)	Empirical range for the Sherwood correlation.

The overall growth rate constant K represents the combined effect of the mass transfer resistance (diffusion through the boundary layer) and the surface integration resistance (incorporation into the crystal lattice). It is calculated via the harmonic mean:

\[ \frac{1}{K} = \frac{1}{k_d} + \frac{1}{k_r} \qquad\Longrightarrow\qquad K = \frac{k_d \cdot k_r}{k_d + k_r} \]

Where k_d is obtained from the Ranz-Marshall correlation and k_r is the experimentally determined surface integration constant. This approach identifies which step limits the overall growth rate.

To maintain precision in the combined resistance model, the following variables should be continuously tracked:

Bulk solute concentration C and saturation concentration C_sat (directly or via temperature and solubility curve).
Crystal size d_p and the slip velocity v between crystal and liquid, which affect the mass transfer coefficient.
Physical properties of the solution: kinematic viscosity ν, diffusivity D_AB, and crystal density ρ_c.
The surface integration constant k_r, which may depend on temperature and impurity levels.

The rate‑limiting step is determined by the relative magnitudes of k_d and k_r. During scale‑up, changes in hydrodynamics can alter this balance:

Increased agitation intensity raises the slip velocity v, thereby increasing k_d; the bottleneck may shift toward surface integration.
Insufficient mixing in large vessels can reduce k_d, making mass transfer the controlling resistance.
Local supersaturation gradients may cause spatial variations in the controlling step, requiring careful analysis of the overall K value.

Worked Example: Overall Crystal Growth Rate for Lactose

A lactose crystal grows in a stirred crystallizer at 40 °C with a bulk concentration of 240.0 kg/m³ and a saturation concentration of 200.0 kg/m³. The crystal diameter is 0.0005 m, the diffusion coefficient is 1.0e-9 m²/s, and the kinematic viscosity is 1.0e-6 m²/s. The slip velocity between crystal and liquid is 0.5 m/s, and the surface integration constant is 2.0e-5 m/s. The crystal density is 1500.0 kg/m³. The following calculation determines the linear growth rate and identifies the rate-limiting step.

Knowns:

\(C_{\text{bulk}} = 240.0\ \text{kg/m}^3\)
\(C_{\text{sat}} = 200.0\ \text{kg/m}^3\)
\(d_p = 5.0 \times 10^{-4}\ \text{m}\)
\(D_{AB} = 1.0 \times 10^{-9}\ \text{m}^2/\text{s}\)
\(\nu = 1.0 \times 10^{-6}\ \text{m}^2/\text{s}\)
\(v = 0.5\ \text{m/s}\)
\(k_r = 2.0 \times 10^{-5}\ \text{m/s}\)
\(\rho_c = 1500.0\ \text{kg/m}^3\)

Step-by-Step Calculation:

Concentration driving force: \(\Delta C = C_{\text{bulk}} - C_{\text{sat}} = 40.0\ \text{kg/m}^3\).
Reynolds number: \(\text{Re} = \frac{d_p \cdot v}{\nu} = 250.0\).
Schmidt number: \(\text{Sc} = \frac{\nu}{D_{AB}} = 1000.0\).
Sherwood number (Ranz-Marshall): \(\text{Sh} = 2.0 + 0.6 \cdot \text{Re}^{0.5} \cdot \text{Sc}^{1/3} = 96.87\).
Mass transfer coefficient: \(k_d = \frac{\text{Sh} \cdot D_{AB}}{d_p} = 1.937 \times 10^{-4}\ \text{m/s}\).
Surface integration constant: Provided as \(k_r = 2.0 \times 10^{-5}\ \text{m/s}\).
Overall growth rate constant: \(\frac{1}{K} = \frac{1}{k_d} + \frac{1}{k_r}\) → \(K = \frac{k_d \cdot k_r}{k_d + k_r} = 1.813 \times 10^{-5}\ \text{m/s}\).
Mass flux: \(R_G = K \cdot \Delta C = 7.25 \times 10^{-4}\ \text{kg/(m}^2\cdot\text{s)}\).
Linear growth rate: \(G = \frac{R_G}{\rho_c} = 4.83 \times 10^{-7}\ \text{m/s}\) (or 0.483 μm/s).
Rate-limiting step: Compare \(k_d\) and \(k_r\): \(k_d = 1.937 \times 10^{-4}\ \text{m/s}\) is larger than \(k_r = 2.0 \times 10^{-5}\ \text{m/s}\), so surface integration is the slower, controlling step.

Final Answer: The linear crystal growth rate is \(G = 4.83 \times 10^{-7}\ \text{m/s}\) (0.483 μm/s). The overall process is limited by surface integration.

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

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