Introduction & Context

The residence time calculation for a continuous crystallizer is a fundamental design tool in process engineering. It determines the volume of the crystallizer required to achieve a specified residence time, which in turn controls nucleation, growth, and product quality. Accurate residence time prediction is essential for scale‑up, energy optimization, and compliance with regulatory specifications in pharmaceutical, food, and chemical manufacturing, and it complements the methods described in batch crystallizer cycle time estimation for batch operations.

Methodology & Formulas

All calculations are expressed in algebraic form. The following variables are used:

  • Qfeed – volumetric feed rate (L h⁻¹)
  • τ – desired residence time (h)
  • T – temperature (°C)
  • η – viscosity (cP)
  • d – reactor diameter (m)
  • L – reactor length (m)
  • ρ – liquid density (kg m⁻³)
  • D – diffusion coefficient (m² s⁻¹)

1. Unit Conversions

  • Feed rate in cubic meters per second:
  • \[ Q_{\text{feed, m}^3\text{/s}} = \frac{Q_{\text{feed, L/h}} \times L_{\text{to m}^3}}{h_{\text{to s}}} \]
  • Reactor volume in cubic meters:
  • \[ V_{\text{m}^3} = \tau \times Q_{\text{feed, L/h}} \times L_{\text{to m}^3} \]

2. Reactor Geometry

  • Cross-sectional area:
  • \[ A_{\text{m}^2} = \frac{\pi d^2}{4} \]
  • Linear velocity (m s⁻¹):
  • \[ u_{\text{m/s}} = \max\!\left(\frac{Q_{\text{feed, m}^3\text{/s}}}{A_{\text{m}^2}},\, 10^{-9}\right) \]

3. Transport Parameters

  • Diffusion coefficient (assumed for lactose at 25 °C):
  • \[ D_{\text{m}^2\text{/s}} = \text{constant value} \]
  • Peclet number:
  • \[ Pe = \frac{u_{\text{m/s}} \, L}{D_{\text{m}^2\text{/s}}} \]

4. Viscosity Conversion

  • Viscosity in Pa s:
  • \[ \mu_{\text{Pa·s}} = \eta \times cP_{\text{to Pa·s}} \]

5. Reynolds Number

  • \[ Re = \frac{\rho \, u_{\text{m/s}} \, d}{\mu_{\text{Pa·s}}} \]

6. Flow Regime Determination

ConditionDescription
\(Re < 1000\)Laminar flow regime
\(Re \ge 1000\)Turbulent flow regime