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.

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