Question

A stirred-tank reactor is to be scaled down from \(10 \mathrm{~m}^{3}\) to \(0.1 \mathrm{~m}^{3}\). The dimensions of the large tank are: \(D_{t}=2 \mathrm{~m} ; D_{i}=0.5 \mathrm{~m} ; N=100 \mathrm{rpm}\). a. Determine the dimensions of the small tank \(\left(D_{p}, D_{i}, H\right)\) by using geometric similarity b. What would be the required rotational speed of the impeller in the small tank if the following criteria were used? 1) Constant tip speed 2) Constant impeller Re number

Short answer

a. The dimensions of the small tank are: D_p = 0.43m, D_i = 0.1075m, H = 0.43m. b. The rotational speeds are 465.3 rpm (constant tip speed) and 2165 rpm (constant Re number).

Step-by-step solution

Calculate the Scale Factor

Determine the scale factor by taking the ratio of the volumes of the small and large tanks. The scale factor (S) is given by \[S = \left( \frac{0.1 \mathrm{~m}^3}{10 \mathrm{~m}^3} \right)^{1/3} = \left( \frac{1}{100} \right)^{1/3} = 0.215\]

Determine the Dimensions of the Small Tank

Using the scale factor, calculate the new dimensions. For geometric similarity: \ \(D_p = 2m \times 0.215 = 0.43m\), \ \(D_i = 0.5m \times 0.215 = 0.1075m\), \ \(H = D_t \times 0.215 = 2m \times 0.215 = 0.43m\)

Calculate the Rotational Speed for Constant Tip Speed

The impeller tip speed is \[V_t = N_t \times D_i \times \frac{\pi}{60} = 100 \space rpm \times 0.5m \times \frac{\pi}{60} = 2.62 \frac{m}{s}\] For the smaller tank, \[N_s \times 0.1075m \times \frac{\pi}{60} = 2.62 \frac{m}{s}\] Solving for the new rotational speed, \[N_s = \frac{2.62 \frac{m}{s} \times 60}{0.1075m \times \pi} = 465.3 \space rpm\]

Calculate the Rotational Speed for Constant Impeller Re Number

The Reynolds number is given by \[Re = \frac{\rho N D_i^2}{\mu}\] For geometric similarity, \[ \frac{\rho N_t D_i^2}{\mu} = \frac{\rho N_s D_i'^2}{\mu} \Rightarrow N_s = N_t \left( \frac{D_i}{D_i'} \right)^2\] Substituting for the values, \[N_s = 100 \space rpm \times \left( \frac{0.5m}{0.1075m} \right)^2 = 100 \space rpm \times 21.65 = 2165 \space rpm\]

Key concepts

Scale Factor Calculation

When scaling down a stirred-tank reactor, the first step is to determine the scale factor. This helps ensure that the smaller tank maintains the same proportion as the larger one. The scale factor, denoted as S, is calculated based on the ratio of the volumes of the small and large tanks.

Given volumes:

• Large tank: 10 m³
• Small tank: 0.1 m³

Using the formula: \[ S = \left( \frac{0.1 \text{ m}^3}{10 \text{ m}^3} \right)^{1/3} = \left( \frac{1}{100} \right)^{1/3} = 0.215 \] This scale factor tells us that each linear dimension of the small tank is 21.5% of the corresponding dimension in the large tank.

Geometric Similarity

Geometric similarity means that the shape and proportions of the smaller tank are the same as those of the larger tank. Once we have the scale factor, we use it to determine the dimensions of the small tank.

For the large tank:

• Diameter of the tank (Dt) = 2 m
• Diameter of the impeller (Di) = 0.5 m
• Height of the tank (H) is assumed to be the same as the tank diameter for convenience = 2 m

Using the scale factor (S = 0.215):

• Small Tank Diameter: \[ D_p = D_t \times S = 2 \text{ m} \times 0.215 = 0.43 \text{ m} \]
  
• Small Impeller Diameter: \[ D_i = D_i \times S = 0.5 \text{ m} \times 0.215 = 0.1075 \text{ m} \]
  
• Small Tank Height: \[ H = D_t \times S = 2 \text{ m} \times 0.215 = 0.43 \text{ m} \]

This ensures that the small reactor has the same geometric proportions as the large reactor.

Rotational Speed Adjustment

When scaling the reactor, the rotational speed of the impeller needs adjustment to maintain certain criteria. There are two main criteria: constant tip speed and constant Reynolds number (Re).

Constant Tip Speed:
The tip speed of the impeller must remain the same in both large and small tanks to ensure similar mixing characteristics. The tip speed is given by: \[ V_t = N_t \times D_i \times \frac{\pi}{60} \] For the large tank, \[ V_t = 100 \text{ rpm} \times 0.5 \text{ m} \times \frac{\pi}{60} = 2.62 \text{ m/s} \] For the small tank, we use: \[ N_s \times 0.1075 \text{ m} \times \frac{\pi}{60} = 2.62 \text{ m/s} \] Solving for the new rotational speed: \[ N_s = \frac{2.62 \text{ m/s} \times 60}{0.1075 \text{ m} \times \pi} = 465.3 \text{ rpm} \]

Constant Impeller Re Number:
To maintain hydrodynamic similarity, the Reynolds number must remain constant. The Reynolds number is given by: \[ Re = \frac{\rho N D_i^2}{\mu} \] For the large tank: \[ Re_t = \frac{\rho N_t D_i^2}{\mu} \] For the small tank, it should be: \[ Re_s = \frac{\rho N_s D_i'^2}{\mu} = Re_t \] This gives the relationship: \[ N_s = N_t \left( \frac{D_i}{D_i'} \right)^2 \] Substituting the values: \[ N_s = 100 \text{ rpm} \times \left( \frac{0.5 \text{ m}}{0.1075 \text{ m}} \right)^2 = 100 \text{ rpm} \times 21.65 = 2165 \text{ rpm} \] Thus, these adjustments ensure that the mixing or agitation quality remains constant during the scaling process.