Batch Adsorption Tank Scale-Up

Reference ID: MET-E316 | Process Engineering Reference Sheets Calculation Guide

Introduction & Context

Batch adsorption tank scale-up is a critical procedure in process engineering, particularly in the pharmaceutical and chemical industries where liquid-phase adsorption is used for purification or recovery. The objective is to translate experimental data from a small-scale laboratory model to a full-scale industrial prototype while maintaining consistent mass transfer performance.

The primary challenge in scale-up is ensuring that the hydrodynamic environment—specifically the turbulence intensity and the liquid‑film mass transfer coefficient—remains predictable. By utilizing the principle of constant power per unit volume (\(P/V\)), engineers can maintain geometric similarity and provide sufficient agitation to overcome external mass transfer resistances at the adsorbent particle surface.

Methodology & Formulas

The scale‑up procedure employs the following relations, where the mass transfer is referenced to the adsorbent particle diameter \(d_{p}\).

1. Fluid and Dimensionless Properties
The Schmidt number (\(\mathrm{Sc}\)) characterizes the ratio of momentum diffusivity to mass diffusivity:

\[ \mathrm{Sc} = \frac{\mu}{\rho \cdot D_{AB}} \]

2. Geometric Similarity and Power Scaling
For geometrically similar tanks where the impeller diameter \(D\) is a fixed fraction of the tank diameter \(T\), the constant power per unit volume criterion dictates the impeller rotational speed \(N\):

\[ N_{M} = N_{P} \cdot \left( \frac{T_{P}}{T_{M}} \right)^{\frac{2}{3}} \]

3. Energy Dissipation and Particle Reynolds Number
The solid-liquid mass transfer in stirred tanks is governed by microscale turbulence, characterized by the specific energy dissipation rate (\(\varepsilon\)). For standard tanks where liquid height equals the tank diameter (\(H = T\)):

\[ \varepsilon = \frac{P}{\rho \cdot V} = \frac{4 \cdot N_{p} \cdot N^{3} \cdot D^{5}}{\pi \cdot T^{3}} \]

The effective slip velocity between turbulent eddies and suspended particles is approximated by Kolmogoroff's theory as \(u_{\text{slip}} \approx (\varepsilon \cdot d_{p})^{1/3}\). The particle Reynolds number (\(\mathrm{Re}_{p}\)) is:

\[ \mathrm{Re}_{p} = \frac{\rho \cdot (\varepsilon \cdot d_{p})^{1/3} \cdot d_{p}}{\mu} \]

4. Mass Transfer Correlation
The Sherwood number (\(\mathrm{Sh}\)) for external mass transfer around spherical particles is correlated by:

\[ \mathrm{Sh} = 2 + 0.6 \cdot \mathrm{Re}_{p}^{\frac{1}{2}} \cdot \mathrm{Sc}^{\frac{1}{3}} \] \[ k = \frac{\mathrm{Sh} \cdot D_{AB}}{d_{p}} \]

Regime	Condition	Applicability
Turbulent particle flow	\(\mathrm{Re}_{p} > 0.1\)	Correlation valid
Schmidt Range	\(0.6 \leq \mathrm{Sc} \leq 2000\)	Correlation valid

To ensure geometric and dynamic similarity during scale-up, process engineers should prioritize the following parameters:

Particle Reynolds number (\(\mathrm{Re}_{p}\)) to maintain flow regime consistency around the adsorbent particles.
Schmidt number (\(\mathrm{Sc}\)) to account for mass transfer properties.
Sherwood number (\(\mathrm{Sh}\)) to correlate the mass transfer coefficient with fluid dynamics.
Power number (\(N_{p}\)) to ensure consistent power input per unit volume across scales.

Maintaining mass transfer efficiency requires balancing the external film resistance with the internal pore diffusion. You should:

Keep the ratio of impeller diameter to tank diameter constant.
Adjust the agitation speed to maintain a constant power input per unit volume (constant \(P/V\)).
Monitor the particle‑phase mass transfer coefficient (\(k\)) to ensure it does not decrease as the boundary layer thickness changes with scale.

Particle size distribution significantly influences the pressure drop and the effective surface area available for adsorption. During scale-up, consider the following:

Ensure the just‑suspended speed (\(N_{js}\)) is met to prevent particle settling at the bottom of the larger tank.
Account for potential attrition of particles caused by higher shear forces near the impeller in larger vessels.
Verify that the ratio of tank diameter to particle diameter remains large enough to avoid wall effects that could bias experimental data.

Batch Adsorption Tank Scale-Up: Constant Power Per Unit Volume

Scenario: A stirred tank adsorption process using a Rushton turbine impeller is to be scaled up from a model tank (diameter 0.3 m) to a prototype tank (diameter 2.0 m). The fluid is water at 20 °C, and the model mass transfer coefficient was measured experimentally. Scale-up is performed using the constant power per unit volume criterion (\(P/V\) = constant) for turbulent mixing in geometrically similar tanks.

Known Inputs & Fluid Properties:

Density, \(\rho = 998.0 \, \mathrm{kg/m^3}\)
Dynamic viscosity, \(\mu = 0.001 \, \mathrm{Pa·s}\)
Molecular diffusivity of adsorbate, \(D_{AB} = 1.0 \times 10^{-9} \, \mathrm{m^2/s}\)
Schmidt number, \(\mathrm{Sc} = \dfrac{\mu}{\rho D_{AB}} = 1002.0\)
Adsorbent particle diameter, \(d_{p} = 1.0 \times 10^{-3} \, \mathrm{m}\) (1 mm)
Tank diameters: prototype \(T_{P} = 2.0 \, \mathrm{m}\), model \(T_{M} = 0.3 \, \mathrm{m}\)
Impeller diameters (1/3 of tank): \(D_{P} = 0.667 \, \mathrm{m}\), \(D_{M} = 0.100 \, \mathrm{m}\)
Prototype impeller speed: \(N_{P} = 150.0 \, \mathrm{rpm}\)
Model mass transfer coefficient (measured): \(k_{M} = 1.50 \times 10^{-4} \, \mathrm{m/s}\)
Power number for Rushton turbine (turbulent): \(N_{p} = 5.0\)

Step-by-Step Calculation:

Convert prototype speed to revolutions per second: \[ N_{P,\mathrm{rps}} = \frac{150.0}{60} = 2.500 \, \mathrm{rps} \]
Compute specific energy dissipation rate for the prototype (\(\varepsilon_P\)): \[ \varepsilon_P = \frac{4 \cdot N_{p} \cdot N_{P,\mathrm{rps}}^3 \cdot D_P^5}{\pi \cdot T_P^3} = \frac{4 \times 5.0 \times (2.500)^3 \times (0.667)^5}{\pi \times (2.0)^3} = 1.646 \, \mathrm{W/kg} \ (\text{or } \mathrm{m^2/s^3}) \]
Prototype particle Reynolds number based on Kolmogoroff eddy slip velocity: \[ \mathrm{Re}_{p,P} = \frac{\rho \cdot (\varepsilon_P \cdot d_p)^{1/3} \cdot d_p}{\mu} = \frac{998.0 \times (1.646 \times 10^{-3})^{1/3} \times 1.0\times10^{-3}}{0.001} = 117.8 \]
Scale-up criterion: constant \(P/V \implies \varepsilon_M = \varepsilon_P\). The required model impeller speed is: \[ N_{M,\mathrm{rps}} = N_{P,\mathrm{rps}} \left( \frac{T_P}{T_M} \right)^{2/3} = 2.500 \times \left( \frac{2.0}{0.3} \right)^{2/3} = 8.850 \, \mathrm{rps} \quad (\text{or } 531.0 \, \mathrm{rpm}) \]
Model particle Reynolds number:
Since \(\varepsilon\) and \(d_p\) are identical at both scales under constant \(P/V\), the Reynolds number is unchanged: \[ \mathrm{Re}_{p,M} = \mathrm{Re}_{p,P} = 117.8 \]
Compute Sherwood number using \(\mathrm{Sh} = 2 + 0.6\, \mathrm{Re}_{p}^{1/2}\, \mathrm{Sc}^{1/3}\): \[ \mathrm{Sc}^{1/3} = (1002.0)^{1/3} = 10.007 \] \[ \mathrm{Sh} = 2 + 0.6 \times (117.8)^{1/2} \times 10.007 = 67.2 \]
Predict mass transfer coefficient from correlation: \[ k_{\mathrm{corr}} = \frac{\mathrm{Sh} \cdot D_{AB}}{d_{p}} = \frac{67.2 \times 1.0\times10^{-9}}{1.0\times10^{-3}} = 6.72 \times 10^{-5} \, \mathrm{m/s} \]
Predict prototype mass transfer coefficient by scaling the model measurement:
Because \(\mathrm{Sh}_P = \mathrm{Sh}_M\) at constant \(P/V\), the scale-up principle guarantees: \[ k_{P,\mathrm{scaled}} = k_M = 1.50 \times 10^{-4} \, \mathrm{m/s} \]

Final Answer: Under the constant \(P/V\) strategy, specific energy dissipation (\(\varepsilon\)) remains equal across scales, yielding a constant particle Reynolds number of 117.8. While the correlation predicts a baseline \(k_{\mathrm{corr}} = 6.72 \times 10^{-5} \, \mathrm{m/s}\), applying the scale-up invariance directly to the experimental data yields \(k_{P,\mathrm{scaled}} = 1.50 \times 10^{-4} \, \mathrm{m/s}\). This demonstrates why constant \(P/V\) is the benchmark method for preserving solid-liquid mass transfer performance during scale-up.

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

"La difficulté attire l'homme de caractère, car c'est en l'étreignant qu'il se réalise." — Charles de Gaulle