Let's break down the problem step by step: **Given Data:** - **Influent Flowrate (Q):** 0.170 m³/s - **Influent BOD₅ (S₀):** 100 mg/L - **Effluent BOD₅ (S):** 15 mg/L - **Mixed Liquor Volatile Suspended Solids (MLVSS, X):** 1,000 mg/L - **Monod Constants:** - \( K_s \) = 60 mg/L (half-saturation constant) - \( \mu_{max} \) = 2.8 d⁻¹ (maximum specific growth rate) - \( k_d \) = 0.05 d⁻¹ (decay rate) - \( Y \) = 0.70 mg MLVSS/mg BOD₅ (yield coefficient) - **RAS flowrate is negligible compared to influent flowrate.** --- ## **Step 1: Convert Flowrate to m³/d** \[ Q = 0.170 \text{ m}^3/\text{s} \times 86,400 \text{ s/d} = 14,688 \text{ m}^3/\text{d} \] --- ## **Step 2: Calculate the Substrate Utilization Rate** \[ Q(S_0 - S) = \text{mass of BOD}_5 \text{ removed per day} \] \[ = 14,688 \text{ m}^3/\text{d} \times (100 - 15) \text{ mg/L} \] \[ = 14,688 \times 85 \text{ mg/L} \] \[ = 1,248,480 \text{ mg/d} \] \[ = 1,248.5 \text{ kg/d} \] --- ## **Step 3: Substrate Balance for Completely Mixed Activated Sludge** The mass balance for substrate in a CMAS reactor is: \[ Q(S_0 - S) = V \cdot r_s \] where \( r_s \) is the substrate utilization rate per unit volume (\( \text{mg/L}\cdot\text{d} \)). The Monod equation for growth rate is: \[ \mu = \mu_{max} \frac{S}{K_s + S} \] The net specific substrate utilization rate: \[ r_s = \frac{1}{Y} \cdot (\mu X) = \frac{1}{Y} \cdot \left( \mu_{max} \frac{S}{K_s + S} X \right) \] So, \[ Q(S_0 - S) = V \cdot \frac{1}{Y} \cdot \left( \mu_{max} \frac{S}{K_s + S} X \right) \] Rearrange for \( V \): \[ V = \frac{Q(S_0 - S) \cdot Y}{\mu_{max} \frac{S}{K_s + S} X} \] --- ## **Step 4: Plug in the Values** **Calculate \(\frac{S}{K_s + S}\):** \[ \frac{S}{K_s + S} = \frac{15}{60 + 15} = \frac{15}{75} = 0.20 \] **Now, substitute all values:** - \( Q(S_0 - S) \) = 1,248,480 mg/d - \( Y \) = 0.70 - \( \mu_{max} \) = 2.8 d⁻¹ - \( \frac{S}{K_s + S} \) = 0.20 - \( X \) = 1,000 mg/L \[ V = \frac{1,248,480 \times 0.70}{2.8 \times 0.20 \times 1,000} \] Calculate denominator: \[ 2.8 \times 0.20 \times 1,000 = 2.8 \times 200 = 560 \] Calculate numerator: \[ 1,248,480 \times 0.70 = 873,936 \] So, \[ V = \frac{873,936}{560} = 1,560.6 \text{ L} \] But this is in **liters** (since Q(S₀-S) was in mg/d and X is mg/L), let's convert to **m³**: \[ V = \frac{873,936}{560,000} = 1.56 \text{ m}^3 \] Wait, this is much too small for a wastewater plant. Let's reexamine the units. --- ### **Check Units Carefully** - \( Q(S_0 - S) \): mg/d - \( X \): mg/L So, \( Q(S_0 - S) \) / \( X \) gives L/d. But in Monod, \( \mu_{max} \frac{S}{K_s + S} \) is d⁻¹. So, \( \frac{Q(S_0 - S)}{\mu_{max} \frac{S}{K_s + S} X} \) gives units of L. Yes, the calculation above is correct in units. But the number is too small for a real aeration tank. Let's double check calculation: - \( Q = 14,688 \) m³/d = 14,688,000 L/d - \( S_0 - S = 85 \) mg/L - So, \( Q(S_0 - S) = 14,688,000 \times 85 = 1,248,480,000 \) mg/d = 1,248.48 kg/d So, numerator is: \[ Q(S_0 - S) \times Y = 1,248,480,000 \times 0.70 = 873,936,000 \text{ mg/d} \] Denominator: \[ \mu_{max} \frac{S}{K_s + S} X = 2.8 \times 0.20 \times 1,000 = 560 \text{ d}^{-1} \text{ mg/L} \] But since X is in mg/L, let's keep the denominator in mg/L \cdot d⁻¹. So, \[ V = \frac{873,936,000 \text{ mg/d}}{560 \text{ mg/L} \cdot \text{d}^{-1}} = 1,560,600 \text{ L} \] \[ = 1,560.6 \text{ m}^3 \] --- ## **Step 5: Final Answer** ### **Required Volume of Aeration Tank:** \[ \boxed{1,560 \text{ m}^3} \] --- ## **Summary Table** | Parameter | Value | |------------------|---------------------| | Influent Flow, Q | 0.170 m³/s (14,688 m³/d) | | Influent BOD₅ | 100 mg/L | | Effluent BOD₅ | 15 mg/L | | MLVSS, X | 1,000 mg/L | | \( K_s \) | 60 mg/L | | \( \mu_{max} \) | 2.8 d⁻¹ | | \( k_d \) | 0.05 d⁻¹ | | \( Y \) | 0.70 | | **Aeration Volume, V** | **1,560 m³** | --- ## **Conclusion** **The required volume of the aeration tank is** \(\boxed{1,560\ \text{m}^3}\) **to meet the effluent BOD₅ limit of 15 mg/L under the given conditions.** --- **If you want to show the step-by-step equations for a report or homework, you can use the above derivation. Let me know if you need calculations for other parameters (e.g., SRT, F/M ratio, etc.)!**