## Question

Fresh ſeed enters a plug-flow activated sludge reactor at a flow rate of 2 mgd (0.088 m/sec) and with a substrate concentration of 300 mg/l. Sludge is recycled from a thickener to the entrance of the reactor with a biomass con. centration of 6000 mg/l. The volume of the plug-flow reactor is 40,000 gal (151 m’). Determine the recycle ratio needed to give a substrate conversion of 95%: The kinetic constants for the reaction are: ko KM – 0.3 hr-1 75 mg/C Y = 0.5 mg/mg COD k0

## Answer

**1. Model development:**

The mass balance equation for the substrate concentration (S) in the reactor is:

dS/dx = Q(Sin – S) – X*r

where:

- Q is the volumetric flow rate (0.088 m³/s)
- Sin is the influent substrate concentration (300 mg/l)
- S is the substrate concentration at any point x along the reactor length
- X is the biomass concentration
- r is the net substrate removal rate

The net substrate removal rate (r) can be expressed using the Monod equation:

r = k₀ * S / (Km + S) – k₀ * X

where:

- k₀ is the decay constant for biomass (0.3 hr⁻¹)
- Km is the half-saturation constant (75 mg/l)

**2. Parameter substitution:**

- Q = 0.088 m³/s
- Sin = 300 mg/l
- V = 151 m³
- X = 6000 mg/l (assuming complete mixing in the recycled sludge)
- k₀ = 0.3 hr⁻¹
- Km = 75 mg/l
- Y = 0.5 mg/mg COD (not used directly in this model, but provides information about biomass yield)
- k₄ = 0 (assumed negligible in this case)

**3. Conversion calculation:**

The substrate conversion (X) is:

X = (Sin – Sout) / Sin

where:

- Sout is the effluent substrate concentration

The desired conversion is 95%, so:

0.95 = (300 mg/l – Sout) / 300 mg/l

Sout = 15 mg/l

**4. Recycle ratio (R):**

The recycle ratio (R) is the ratio of the recycled sludge flow rate (Qr) to the fresh feed flow rate (Q):

R = Qr / Q

We can express Qr as a function of the reactor effluent and target effluent concentrations:

Qr = Q * (Sout – Sin) / X

Substituting all the values:

Qr = 0.088 m³/s * (15 mg/l – 300 mg/l) / 6000 mg/l

Qr = -0.0386 m³/s

(Note: the negative value indicates a net flow out of the reactor due to the high recycle ratio and low effluent concentration.)

Therefore, the actual recycle flow rate cannot be negative. This suggests that it might not be possible to achieve a 95% conversion with the given parameters within the specified reactor volume. However, one could:

- Reduce the desired conversion threshold.
- Increase the reactor volume to allow for lower recycle ratios and higher conversions.
- Modify the model parameters (e.g., increase Y) to improve substrate removal efficiency.

Further information or assumptions regarding operating conditions might be needed to refine the analysis and obtain a realistic solution for the recycle ratio.