# Optimized Analytical-Numerical Procedure for Ultrasonic Sludge Treatment for Agricultural Use
## 5 Temperature-Humidity Coupled Model for Sludge Treatment  
### Quantification of the Acoustic Power  
Let \(Oxyz\) be an ortho-normal Cartesian coordinate system where \(\Omega\subset\mathbb{R}^{3}\) represents the sludge; then, the generic vector \(\mathbf{x}=(x,y,z)\subset\Omega\) represents a point on the sludge. The acoustic energy, \(E\), generated and transferred to the sludge sample during US sonication (following pretreatment in an oven at the initial temperature of 373 \({}^{\circ}\)C for two hours) is transformed, in part, into heat that produces an increase in temperature \(T(\mathbf{x},t)\) that can be  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Heavy Metals (mg/kg)** & **National Decree** & **Sludge Analysis of the Plant** \\
**and Bacteria (MPN/gSS)** & **DL99—29 January 1992** & **Sludge Analysis of the Plant** \\ \hline arsenic & n.d. & // \\ copper & 1000 & 700–800 \\ zinc & 2500 & 700–1200 \\ cadmium & 20 & \textless{}1 \\ mercury & 10 & \textless{}1 \\ lead & 750 & 90–160 \\ nickel & 300 & 50–90 \\ organic carbon (\% sludge) & 20 (minimum) & 25–30 \\ phosphorus (\% sludge) & 20 (minimum) & 25–30 \\ nitrogen (\% sludge) & 1.5 (minimum) & 4.5–5.5 \\ salmonella & 103 (maximum) & 30–60 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison between the parameters given in DL99 for sludge use in agriculture and the data collected on the sludge produced by the plant.  
Figure 3: The experimental apparatus for sludge treatment process by ultrasonic sonication. **Left:** the amplifying device with current values increasing in percentage from 0 to 60 A. **Right**: the tank containing the sludge during ultrasonic sonication.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Dimensions & 445 \(\times\) 545 \(\times\) 1560 mm \\ Electrical supply & 230 \(\%\) 50 Hz \\ US frequency & 50 kHz, constant \\ Tank volume & 8 L \\ Maximum input flow rate & 3000 L/h, i.e., 50 L/min \\ Contact time & 2.40 s per every sludge sample \\ Sonication time & 0–150 min, sampling time 5 min \\ \hline \hline \end{tabular}
\end{table}
Table 3: Operating parameter of sludge sonication treatment.  
quantified using a calorimetric approach (highlighting the direct proportionality between \(T(\mathbf{x},t)\) and \(P\)). The idea is based on the thermodynamic principle that \(P\) supplied by the US is converted into heat, and thus, the acoustic power can be calculated by measuring the temperature rise in the sludge.  
The US energy (expressed in J) that causes an increase in \(T(\mathbf{x},t)\) can be expressed as [61; 62; 63]:  
\[Q=MC_{p}\Delta T(\mathbf{x},t) \tag{1}\]  
where \(Q\) and \(M\) are the amount of heat absorbed by the slurry and its mass \(M\) (kg); \(C_{p}\) is the specific heat capacity (J/kg K); and \(\Delta T(\mathbf{x},t)\) is the increment of \(T(\mathbf{x},t)\) (K) (experimentally measurable by sensors). Then, the power \(P\), if \(\Delta t\) is the time interval during which heat transfer occurred, takes the form:  
\[P=\frac{MC_{p}\Delta T(\mathbf{x},t)}{\Delta t}, \tag{2}\]  
which, in an infinitesimal time frame, becomes:  
\[P=MC_{p}\frac{dT(\mathbf{x},t)}{dt}. \tag{3}\]