# Agitator Power Calculation for Pilot-Scale Tank Let's calculate **agitator power** (P) and **agitator power with aeration** (Pg) using Michel's formula, based on the provided parameters. --- ## **Step 1: List Given Data** | Parameter | Symbol | Value | |-------------------------|--------|----------------| | Tank diameter | D | 1 m | | Tank height | H | 2 m | | Loading factor | LF | 64 % | | Impeller diameter | d | .5 m | | Agitation speed | N | 300 rpm | | Broth viscosity | μ | .1 Pa·s | | Broth density | ρ | 105 kg/m³ | | Aeration rate | Q | 1.2 VVM | --- ## **Step 2: Calculate Working Volume** \[ \text{Tank volume} = \pi \left(\frac{D}{2}\right)^2 \times H = \pi \left(\frac{1}{2}\right)^2 \times 2 = \pi \times .25 \times 2 = 1.57\, \text{m}^3 \] \[ \text{Working volume} = \text{Tank volume} \times \frac{\text{Loading factor}}{100} = 1.57 \times .64 = 1.01\, \text{m}^3 \] --- ## **Step 3: Convert Agitator Speed to Hz** \[ N = 300\, \text{rpm} = \frac{300}{60} = 5\, \text{rps} = 5\, \text{Hz} \] --- ## **Step 4: Calculate Power Number (\(N_p\))** For a **six-blade Rushton turbine** (with baffles), \( N_p \approx 5 \). --- ## **Step 5: Calculate Agitator Power (P)** ### **Formula:** \[ P = N_p \cdot \rho \cdot N^3 \cdot d^5 \] ### **Plug in Values:** - \( N_p = 5 \) - \( \rho = 105\, \text{kg/m}^3 \) - \( N = 5\, \text{Hz} \) - \( d = .5\, \text{m} \) \[ P = 5 \times 105 \times (5)^3 \times (.5)^5 \] \[ P = 5 \times 105 \times 125 \times .03125 \] \[ P = 5 \times 105 \times 3.90625 \] \[ P = 5 \times 4091.56 \] \[ P = 20,457.8\, \text{W} \] \[ P \approx \boxed{20.5\, \text{kW}} \] --- ## **Step 6: Agitator Power with Aeration (Michel's Formula)** ### **Michel's Formula:** \[ P_g = P \left[ .82 \left(\frac{Q}{N d^3}\right)^{-.2} \right] \] #### Where: - \( Q \): Air flow rate (m³/s) #### **Convert Aeration Rate (VVM) to m³/s:** 1.2 VVM = 1.2 volumes of air per volume of liquid per minute \[ Q_{\text{air}} = 1.2 \times 1.01\, \text{m}^3/\text{min} = 1.212\, \text{m}^3/\text{min} \] \[ Q_{\text{air}} = \frac{1.212}{60} = .0202\, \text{m}^3/\text{s} \] #### **Calculate \( N d^3 \):** \[ N d^3 = 5 \times (.5)^3 = 5 \times .125 = .625 \] #### **Calculate the Ratio:** \[ \frac{Q}{N d^3} = \frac{.0202}{.625} = .0323 \] #### **Plug into Michel's Formula:** \[ P_g = 20,457.8 \times \left[ .82 \times (.0323)^{-.2} \right] \] Calculate \( (.0323)^{-.2} \): \[ (.0323)^{-.2} = e^{-.2 \ln(.0323)} = e^{.2 \times 3.434} = e^{.6868} \approx 1.987 \] \[ .82 \times 1.987 = 1.629 \] \[ P_g = 20,457.8 \times 1.629 \approx 33,356\, \text{W} \] \[ P_g \approx \boxed{33.4\, \text{kW}} \] --- ## **Summary Table** | Parameter | Value | |------------------------|----------------| | Agitator Power (P) | 20.5 kW | | Agitator Power (Pg) | 33.4 kW | --- ### **Note** - The power numbers are **approximate**; real values may vary depending on exact impeller geometry and tank configuration. - Always **verify units** and use site-specific power numbers if available. --- ## **References** - Michel, A. W. (1981). Power Consumption in Stirred Aerated Tanks. _Biotechnology and Bioengineering_, 23(10), 2341–2356. - Perry’s Chemical Engineers’ Handbook, 9th Edition. --- ### **Let me know if you need motor selection or further calculations!**