Let's solve **each question step-by-step** using the data provided. --- # (5) **Downstream Filtration** ## **Given Data** - **Tank diameter (D):** 1 m - **Tank height (H):** 2 m - **Loading factor:** 64% - **Batch volume = Tank volume × Loading factor** - **Operation time for filtration:** 20 h - **Cycle time per filter press:** 2 h - **Ratio of wet residue/filter cake to feed liquid volume:** 30% (i.e., for every 1 L of feed, .3 L of wet cake is produced) - **Filling coefficient inside filter frame:** .8 - **Model BAS20/635-25:** total internal frame volume V = 260 L (per press, per cycle) --- ## **Step 1: Calculate the Batch Feed Volume** ### **Fermenter Working Volume** \[ \text{Fermenter Volume} = \pi r^2 h = \pi \left(\frac{1}{2}\right)^2 \times 2 = \pi \times .25 \times 2 = \pi \times .5 \approx 1.57~\text{m}^3 \] \[ \text{Loading Factor} = 64\% \] \[ \text{Working Volume} = 1.57 \times .64 \approx 1.01~\text{m}^3 = 101~\text{L} \] --- ## **Step 2: Adjust for Wet Filter Cake** \[ \text{Let } V = \text{feed liquid volume to filter} \] \[ \text{Wet Residue Volume} = .3V \] \[ \text{Total Volume to Fill Filter Frame} = \text{Wet Residue Volume} \] --- ## **Step 3: Filter Press Frame Utilization per Cycle** \[ \text{Effective Filter Frame Volume per Cycle (per press)} = .8 \times 260~\text{L} = 208~\text{L} \] --- ## **Step 4: Number of Cycles Needed** \[ \text{Wet filter cake to process per batch} = .3 \times 101 = 303~\text{L} \] Each press can process **208 L per cycle** (of cake). \[ \text{Number of cycles required (per press)} = \frac{303}{208} \approx 1.46 \] But, **each press cycle is 2 hours**, and the operation must be completed in **20 hours**. \[ \text{Total cycles possible per press in 20 hours} = \frac{20}{2} = 10 \] So, **one filter press can run 10 cycles in 20 hours**, processing: \[ 208~\text{L/cycle} \times 10~\text{cycles} = 208~\text{L} \] This is **much more** than required. But, the process must be completed in 20 hours, and the calculation above is for one batch (303 L of cake per batch). \[ \text{Number of presses required} = \frac{1.46}{10} = .146 \] But this value is **less than 1**, meaning **1 plate-and-frame filter press is sufficient**. --- ## **Final Answer for (5):** ### **Number of Plate-and-Frame Filter Presses Required** \[ \boxed{1} \] **One BAS20/635-25 filter press is sufficient to complete the batch filtration within 20 hours.** --- # (6) **Downstream Ion-Exchange Purification** ## **Given Data** - **Pilot column:** Height = 1. m, Diameter = .3 m - **Pilot flow rate:** 1.2 m³/h - **Production flow rate:** 2.7 m³/h - **Production step must finish in:** 20 h - **Maintain same linear velocity (\( \omega \))** --- ## **Step 1: Calculate Linear Velocity in Pilot System** \[ \text{Column cross-sectional area} = A = \pi r^2 = \pi \left(\frac{.3}{2}\right)^2 = \pi (.15)^2 \approx .0707~\text{m}^2 \] \[ \text{Linear velocity } (\omega) = \frac{\text{Flow rate}}{\text{Area}} = \frac{1.2}{.0707} \approx 16.97~\text{m/h} \] --- ## **Step 2: Calculate Required Area for Production Scale** \[ \text{Required area} = \frac{\text{Production flow rate}}{\omega} = \frac{2.7}{16.97} \approx .1592~\text{m}^2 \] --- ## **Step 3: Choose Scaled Bed Height** Pilot bed height = 1. m. To finish in 20 h, the **bed volume must be sufficient for the batch**. \[ \text{Batch volume to treat} = 101~\text{L} = 1.01~\text{m}^3 \] But, since flow is 2.7 m³/h, in 20 h: \[ \text{Total processed volume} = 2.7 \times 20 = 54~\text{m}^3 \] But only **1.01 m³** is actually needed. Since we're told to maintain linear velocity and scale area, we keep **height = 1. m**. --- ## **Step 4: Calculate Required Diameter** \[ A = \pi \left(\frac{d}{2}\right)^2 = .1592~\text{m}^2 \] \[ \frac{d}{2} = \sqrt{\frac{.1592}{\pi}} = \sqrt{.0507} \approx .225~\text{m} \] \[ d = 2 \times .225 = .45~\text{m} \] --- ## **Final Answer for (6):** ### **Scaled Ion-Exchange Bed Dimensions** - **Height:** \(\boxed{1.~\text{m}}\) - **Diameter:** \(\boxed{.45~\text{m}}\) --- ## **Summary Table** | Part | Final Answer | |------|--------------| | (5) | 1 BAS20/635-25 plate-and-frame filter press required | | (6) | Ion-exchange bed, height = 1. m, diameter = .45 m | --- If you need full workings or clarification, let me know!