Introduction & Context

The calculation presented here determines the required baffle width (W_b) and evaluates the impact of baffles on the power consumption of a stirred-tank reactor. Proper baffle sizing is essential for:

Preventing vortex formation and ensuring adequate axial mixing.
Increasing the turbulent shear rate to improve mass and heat transfer.
Predicting the additional motor load introduced by the presence of baffles.

This methodology is routinely applied in process-engineering design of batch, fed-batch, and continuous reactors where a standard multi-baffle configuration is used.

Methodology & Formulas

Geometric Parameters

Given the tank diameter T and the chosen baffle-to-tank ratio b, the baffle width is obtained from:

\[ W_{b} = b\,T \]

The bottom clearance is typically taken as a fraction c of the baffle width:

\[ \text{clearance} = c\,W_{b} \]

Impeller Kinematics

The impeller tip speed (U_tip) follows from the impeller diameter D and rotational speed N (revolutions per second):

\[ U_{\text{tip}} = \pi D N \]

Hydrodynamic Regime

The Reynolds number based on impeller dimensions is:

\[ Re = \frac{\rho\,N\,D^{2}}{\mu} \]

where ρ is fluid density and μ is dynamic viscosity. The flow regime is assessed against critical Reynolds numbers (Re_lam for laminar and Re_turb for turbulent). The criteria are presented in the table below.

Condition	Flow Regime
\(Re < Re_{lam}\)	Laminar (viscous-dominated)
\(Re_{lam} \le Re \le Re_{turb}\)	Transitional
\(Re > Re_{turb}\)	Turbulent (inertia-dominated)

Power Consumption

The dimensionless power number (N_p) relates the shaft power to fluid properties and impeller geometry. For an unbaffled tank, the power number is denoted N_p,un, and the presence of baffles increases it by a factor k_b:

\[ N_{p,b} = k_{b}\,N_{p,un} \]

The shaft power (P) in watts is then calculated as:

\[ P = N_{p}\,\rho\,N^{3}\,D^{5} \]

Separate expressions are used for the unbaffled and baffled cases:

\[ P_{un} = N_{p,un}\,\rho\,N^{3}\,D^{5} \] \[ P_{b} = N_{p,b}\,\rho\,N^{3}\,D^{5} = k_{b}\,N_{p,un}\,\rho\,N^{3}\,D^{5} \]

The power increase factor caused by the baffles is therefore:

\[ \text{Power\_ratio} = \frac{P_{b}}{P_{un}} \]

Summary of Variables

T – Tank diameter (m)
H – Liquid height (m)
D – Impeller diameter (m)
N – Impeller speed (rev s^-1)
ρ – Fluid density (kg m^-3)
μ – Dynamic viscosity (Pa·s)
b – Baffle-to-tank width ratio (dimensionless)
n_b – Number of baffles (integer)
c – Clearance-to-baffle-width fraction (dimensionless)
N_p,un – Power number for the unbaffled configuration (dimensionless)
k_b – Empirical multiplier accounting for baffle effect (dimensionless)
Re_lam, Re_turb – Critical Reynolds numbers defining flow regimes (dimensionless)

The optimal baffle width is typically based on the tank diameter and the desired flow pattern. Follow these steps:

Measure the internal diameter (D) of the tank.
Choose a width that is 1/3 to 1/2 of D. A common starting point is 0.33 × D.
Check the clearance between the baffle edge and the tank wall; maintain at least 5 mm to avoid dead zones.
Validate the selected width with CFD or pilot-scale testing to ensure adequate mixing.

Several key variables influence the baffle count:

Tank geometry: Cylindrical tanks usually need 4–6 baffles; rectangular tanks may require fewer.
Viscosity of the fluid: Higher viscosity often demands more baffles to break up vortex formation.
Impeller type and size: Large or high-speed impellers generate stronger circulation, potentially reducing the needed baffles.
Process requirements: Desired mixing time, shear rate, and gas dispersion dictate baffle density.

Fluid velocity interacts with baffle spacing in two main ways:

Higher velocities increase shear, allowing wider spacing without loss of mixing efficiency.
At low velocities, tighter spacing (shorter distance between baffles) helps prevent vortex formation and ensures uniform flow.

As a rule of thumb, keep the distance between adjacent baffles between 0.5 × D and 1.0 × D, adjusting based on observed velocity profiles.

Standard dimensions provide a convenient starting point, but they are not universally optimal. Consider the following before adopting a standard size:

Specific process chemistry (corrosiveness, temperature) may require custom materials or thickness.
Scale-up or scale-down projects often need dimension adjustments to maintain similar Reynolds numbers.
Regulatory or safety constraints might limit baffle size in certain industries.

Perform a brief engineering review or CFD analysis to confirm that the standard size meets performance criteria for each new application.

Worked Example – Sizing Baffles for a Small-Scale Process Mixer

A specialty-chemical plant needs to scale-up a blending operation from pilot data. The existing 1.5 m diameter, flat-bottom vessel will be retro-fitted with wall baffles to improve heat-transfer and reduce vortexing. The duty requires the impeller power to rise from the current un-baffled level of 0.095 kW to about 0.24 kW while keeping the same 90 rpm shaft speed. Determine the baffle width and the number of baffles required.

Knowns

Tank diameter T = 1.5 m
Liquid height H = 1.5 m
Impeller diameter D = 0.5 m
Rotational speed N = 90 rpm = 1.5 rps
Liquid density ρ = 1000 kg m^-3
Liquid viscosity μ = 0.001 Pa·s
Target power rise factor ≈ 2.5
Standard baffle ratio W_b/T = 1/11 (industry rule-of-thumb)

Step-by-Step Calculation

Compute tip speed U_tip: \[ U_{tip} = \pi D N = \pi (0.5)(1.5) = 2.356 \text{ m s}^{-1} \]
Calculate Reynolds number: \[ Re = \frac{\rho N D^2}{\mu} = \frac{1000 \times 1.5 \times 0.5^2}{0.001} = 375\,000 \] Flow is fully turbulent.
Un-baffled power number (vendor curve) N_p,unbaffled = 0.9: \[ P_{unbaffled} = N_{p,unbaffled} \rho N^3 D^5 = 0.9 \times 1000 \times 1.5^3 \times 0.5^5 = 94.9 \text{ W} \approx 0.095 \text{ kW} \]
Required power with baffles: \[ P_{baffled} = 2.5 \times P_{unbaffled} = 2.5 \times 94.9 = 237 \text{ W} \approx 0.237 \text{ kW} \]
Implied baffled power number: \[ N_{p,baffled} = \frac{P_{baffled}}{\rho N^3 D^5} = \frac{237}{1000 \times 1.5^3 \times 0.5^5} = 2.25 \] This agrees with literature for a 4-baffle configuration.
Choose 4 baffles (standard). Baffle width: \[ W_b = \frac{T}{11} = \frac{1.5}{11} = 0.136 \text{ m} = 136 \text{ mm} \]
Clearance between baffle and wall: \[ c = 0.02 T = 0.02 \times 1.5 = 0.027 \text{ m} = 27 \text{ mm} \]

Final Answer

Install 4 baffles, each 136 mm wide, set 27 mm off the tank wall. This yields a baffled power draw of 0.24 kW at 90 rpm, meeting the process requirement.

"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