Let's solve the problem step-by-step as per the requirements: --- # Step 1: **Calculate Agitation Power (P) for Pilot-Scale Tank** ### **Given Data** - Tank diameter, \( D = 1 \) m - Tank height, \( H = 2 \) m - Impeller diameter, \( d = .5 \) m - Agitation speed, \( N = 300 \) rpm \( = 300/60 = 5 \) s\(^{-1}\) - Broth viscosity, \( \mu = .1 \) Pa·s - Broth density, \( \rho = 105 \) kg/m³ ### **Assumptions** - Four baffles, two-layer six-blade turbine (typical for bioreactors) - Power number (\(N_P\)) for a standard Rushton turbine is about 5 (for fully baffled tanks). #### **Convert agitation speed to revolutions per second:** \[ N = \frac{300}{60} = 5 \ \text{s}^{-1} \] ### **Power Required (No Aeration):** \[ P = N_P \cdot \rho \cdot N^3 \cdot d^5 \] Where: - \(N_P = 5\) - \(\rho = 105\) kg/m³ - \(N = 5\) s\(^{-1}\) - \(d = .5\) m \[ P = 5 \times 105 \times (5)^3 \times (.5)^5 \] \[ (5)^3 = 125, \quad (.5)^5 = .03125 \] \[ P = 5 \times 105 \times 125 \times .03125 \] \[ = 5 \times 105 \times 3.90625 \] \[ = 5 \times 4091.56 = 20,457.8 \ \text{W} \approx 20.46 \ \text{kW} \] --- ### **With Aeration (Michel’s Formula):** Michel’s correlation for power reduction due to aeration: \[ P_g = P \left[1 - .15 \left(\frac{Q}{N d^3}\right)^{.56}\right] \] Where: - \(Q\) = volumetric flow rate of air (m³/s) #### **Calculate \(Q\) from VVM:** VVM = 1.2 (volume of air per volume of liquid per minute) - Working volume (\(V\)) = Loading factor \(\times\) Tank volume Loading factor = 64% Tank volume (\(V_{tank}\)) = \(\pi/4 \times D^2 \times H = \pi/4 \times 1^2 \times 2 = 1.5708\) m³ Working volume = \(.64 \times 1.5708 = 1.0053\) m³ - Air flow per min = \(1.2 \times 1.0053 = 1.206\) m³/min Per second: \(Q = \frac{1.206}{60} = .0201\) m³/s #### **Calculate \(Q / (N d^3)\):** \[ N d^3 = 5 \times (.5)^3 = 5 \times .125 = .625 \] \[ \frac{Q}{N d^3} = \frac{.0201}{.625} = .0322 \] \[ (.0322)^{.56} = .156 \] \[ P_g = 20.46 \times [1 - .15 \times .156] = 20.46 \times [1 - .0234] \] \[ P_g = 20.46 \times .9766 = 19.98 \ \text{kW} \] --- ## **Summary Table for (3)** | Power Type | Value (kW) | |-----------------|-----------| | Without aeration (\(P\)) | 20.46 | | With aeration (\(P_g\)) | 19.98 | --- # Step 2: **Scale-Up Calculation** ### **Annual Production Target:** - Lipase required = 30,000 kg/year - Working days/year = 300 - Daily production target = \( \frac{30,000}{300} = 100 \) kg/day ### **Per-batch lipase produced (pilot):** - Final concentration = 18.18 g/L = .01818 kg/L - Working volume = 1.0053 m³ = 1005.3 L \[ \text{Lipase produced per batch} = 18.18 \ \frac{g}{L} \times 1005.3 \ L = 18,287 \ g = 18.29 \ kg \] - **Fermentation time**: 180 h = 7.5 days + 60 h (operation) = 10 days/cycle \[ \text{Batches per year (pilot)} = \frac{300}{10} = 30 \] \[ \text{Total annual lipase (pilot)} = 18.29 \times 30 = 548.7 \ \text{kg/year} \] ### **Scale-up Factor:** \[ \text{Scale-up factor} = \frac{30,000}{548.7} = 54.66 \] ### **Required Production-scale Bioreactor Volume:** \[ \text{Production scale volume} = 1.0053 \times 54.66 = 54.94 \ \text{m}^3 \] --- ## **Summary Table for (4)** | Parameter | Value | |-----------------------------------|-----------| | Production-scale volume | 54.94 m³ | | Scale-up factor | 54.66 | --- # Step 3: **Production Tank Main Dimensions, Linear Velocity of Air, Rotational Speed, and Impeller Power** ### **A. Tank Dimensions** Assume same proportions (height/diameter = 2): - Let \(D_p\) = production tank diameter, \(H_p = 2D_p\): \[ V = \frac{\pi}{4} D_p^2 H_p \text{ (ignore headspace for design)} \] \[ 54.94 = \frac{\pi}{4} D_p^2 (2 D_p) = \frac{\pi}{2} D_p^3 \] \[ D_p^3 = \frac{54.94}{\pi/2} = \frac{54.94 \times 2}{\pi} = \frac{109.88}{3.1416} = 34.97 \] \[ D_p = (34.97)^{1/3} = 3.28 \ \text{m} \] \[ H_p = 2 \times 3.28 = 6.56 \ \text{m} \] ### **B. Impeller Diameter** Keep same ratio as pilot: \[ \frac{d}{D} = \frac{.5}{1} = .5 \implies d_p = .5 \times 3.28 = 1.64 \ \text{m} \] ### **C. Rotational Speed (Constant Tip Speed)** Tip speed \(u = \pi d N\) Keep same tip speed as pilot: \[ u_{pilot} = \pi \times .5 \times 5 = 7.85 \ \text{m/s} \] \[ N_p = \frac{u_{pilot}}{\pi d_p} = \frac{7.85}{\pi \times 1.64} = \frac{7.85}{5.153} = 1.52 \ \text{s}^{-1} \] \[ \text{rpm} = 1.52 \times 60 = 91.2 \ \text{rpm} \] ### **D. Linear Air Velocity** Use same VVM: \[ Q_p = V_p \times 1.2 = 54.94 \times 1.2 = 65.93 \ \text{m}^3/\text{min} \] \[ Q_p = \frac{65.93}{60} = 1.099 \ \text{m}^3/\text{s} \] Superficial velocity (\(u_s\)): \[ u_s = \frac{Q_p}{\frac{\pi}{4} D_p^2} = \frac{1.099}{\frac{\pi}{4} \times (3.28)^2} = \frac{1.099}{8.454} = .13 \ \text{m/s} \] ### **E. Impeller Power (Production Scale)** \[ P_p = N_P \rho N_p^3 d_p^5 \] \[ N_p = 1.52 \ \text{s}^{-1} \quad d_p = 1.64 \ \text{m} \] \[ (1.64)^5 = 11.51 \] \[ (1.52)^3 = 3.51 \] \[ P_p = 5 \times 105 \times 3.51 \times 11.51 = 5 \times 105 \times 40.41 \] \[ = 5 \times 42,430.5 = 212,152.5 \ \text{W} \approx 212.2 \ \text{kW} \] With aeration (using same Michel’s factor as pilot, approx .9766), \[ P_{g,p} = 212.2 \times .9766 = 207.2 \ \text{kW} \] --- ## **Final Summary Table: Production Tank** | Parameter | Value | |---------------------------------|-------------| | Volume (\(V_p\)) | 54.94 m³ | | Diameter (\(D_p\)) | 3.28 m | | Height (\(H_p\)) | 6.56 m | | Impeller diameter (\(d_p\)) | 1.64 m | | Rotational speed (\(N_p\)) | 91.2 rpm | | Linear air velocity (\(u_s\)) | .13 m/s | | Impeller power (\(P_p\)) | 212.2 kW | | With aeration (\(P_{g,p}\)) | 207.2 kW | --- # **Final Answers** ### **(3) Agitator Power (Pilot Tank)** - **Without aeration (\(P\)):** **20.46 kW** - **With aeration (\(P_g\)):** **19.98 kW** ### **(4) Scale-up Calculation** - **Production-scale reactor volume:** **54.94 m³** - **Scale-up factor:** **54.66** - **Production tank main dimensions:** - **Diameter:** 3.28 m - **Height:** 6.56 m - **Impeller diameter:** 1.64 m - **Rotational speed:** 91.2 rpm - **Linear air velocity:** .13 m/s - **Impeller power:** 212.2 kW (without aeration), **207.2 kW** (with aeration) --- If you need any derivations or further breakdown for a specific calculation, let me know!