Let's break down the problem into its required steps: --- ## **Step 1: Power Spectral Density (PSD) Relationship** The given governing relationship for the Power Spectral Density (PSD) of the effluent substrate concentration is: \[ S_{yy}(f) = \frac{S_{xx}(f)}{[1 + (2\pi f \theta)^2][1 + (k\theta/(1 + k\theta))^2]} \] where: - \( S_{yy}(f) \): PSD of the effluent substrate concentration - \( S_{xx}(f) \): PSD of the influent substrate concentration - \( \theta \): system time constant - \( k \): process gain (unitless; not given directly, but let’s assume \( k \) is incorporated or its value is not required for this step) - \( f \): frequency --- ## **Step 2: Calculate Effluent Power at .05 Hz** Given: - Influent variance, \( \sigma_{x}^2 = 250. \) - System time constant, \( \theta = 2. \) hours - Frequency, \( f = .05 \) Hz Assumptions: - The influent process is white noise, so \( S_{xx}(f) = \sigma_{x}^2 \) - For simplicity, let’s assume \( k \) is much smaller than \( 1/\theta \), so the denominator simplifies to \( [1 + (2\pi f\theta)^2]^2 \) Now, plug in the values: \[ S_{yy}(f) = \frac{250.}{\left[1 + (2\pi \times .05 \times 2.)^2\right]^2} \] Calculate the denominator: - \( 2\pi \times .05 \times 2. = .628 \) - \( (.628)^2 = .394 \) - \( 1 + .394 = 1.394 \) - \( (1.394)^2 = 1.944 \) So, \[ S_{yy}(.05) = \frac{250.}{1.944} \approx 128.6 \] --- ## **Step 3: Final Answer** \[ \boxed{ S_{yy}(.05) \approx 128.6 } \] This is the effluent power at .05 Hz. --- ## **Step 4: Analytical Framework for Cross-Correlation \(R_{sh}(\tau)\)** - **Variables:** - \( \tau_s(t) \): Hydraulic shear stress (input) - **Sloughing events**: Sudden loss of biomass from the filter surface (output) - **Cross-correlation \( R_{sh}(\tau) \)**: Measures the relationship between shear stress and sloughing as a function of time lag \( \tau \). - **Approach:** - Model \( \tau_s(t) \) as a random process. - Define sloughing events as a response. - Compute \( R_{sh}(\tau) = E[\tau_s(t) \cdot \text{slough}(t+\tau)] \). --- ## **Step 5: Energy Spectral Density Identity for Sloughing** \[ P(\text{slough}) = \int_^\infty S_{\tau\tau}(f) \cdot W(f) \, df \] - \( S_{\tau\tau}(f) \): Energy spectral density of shear stress - \( W(f) \): Mechanical failure transfer function, mapping turbulence to sloughing energy --- ## **Step 6: Governing Logic for Filter Run Time** - **Stochastic-clogging rate**: Modeled as a function of the effluent PSD and sloughing statistics. - **Filter run time**: Minimized when the integral of the clogging rate over time reaches a threshold. - **Optimization**: Use the derived PSD to schedule backwash cycles to maximize filter run time and minimize downtime. --- ## **Summary Table** | **Step** | **Result** | |--------------------------|-----------------------------| | Effluent power at .05 Hz| \( \approx 128.6 \) | | Cross-correlation | \( R_{sh}(\tau) = E[\tau_s(t) \cdot \text{slough}(t+\tau)] \) | | Energy Spectral Identity | \( P(\text{slough}) = \int S_{\tau\tau}(f) W(f) df \) | | Filter run logic | Use PSD & sloughing to optimize backwash schedule | --- ### **Final Answer** \[ \boxed{S_{yy}(.05) \approx 128.6} \] with detailed analytical framework for the cross-correlation and energy mapping provided above.