Sonochemical Effect Induced by Hydrodynamic Cavitation

11 Energy Partition, Reactor Integration, and Scale Up

Only a fraction of the initial pump or shaft energy is ultimately converted into chemical products. The majority of input energy is dissipated through various physical and chemical pathways, including:

• Fluid dynamics: Ordinary turbulence, viscous losses, and large scale fluid circulation

• Thermodynamic and acoustic effects: Thermal heating and pressure waves

• Cavitation phenomena: Bubble rebound, microjets, and material erosion

Didenko and Suslick [26] quantified the energy partition among photons, radicals, ions, and mechanical outputs for single bubble acoustic cavitation, demonstrating that chemical energy conversion represents a minor yet measurable branch.

Because hydrodynamic reactors introduce significant fluid circulation and permanent pressure losses, precise definition of the system boundary remains essential for accurate performance evaluation and scale up.

Figure 11.1. Illustrative partition of collapse energy among shock waves, rebound, microjets and shear, thermal energy, chemical products, and light [81].

Chemical energy efficiency ($\eta$) can be formally quantified as:

where $n_j$ is moles of chemical product j, $\mathrm{\Delta }{G}_j$ is a selected Gibbs energy value for its formation or useful reaction, and $E_{\mathrm{input}}$ is the stated energy boundary.

For repeated pass treatment with a constant per pass removal fraction $f_p$, a simple model is:

where N is the number of passes. The model is useful for early design but fails when fluid properties, intermediates, or radical demand change with conversion.

A continuous reactor with residence time distribution E(t) gives outlet concentration:

for a first order reaction with rate constant k.

For geometrically similar devices, a pressure drop similarity criterion can be written:

where Eu is the Euler number. Maintaining Eu alone does not guarantee similar cavitation chemistry because nuclei, Reynolds number, residence time, vapor pressure, and gas content may differ.

A useful collapse exposure density is:

where $N_{\mathrm{active}}$ is the number of chemically active collapse events, $E_{\mathrm{collapse}}$ is mean energy per active collapse, and $V_{\mathrm{treated}}$ is treated volume. $D_c$ is conceptual because the two microscopic quantities are difficult to measure, but it identifies the scale up target more directly than void fraction.

Figure 11.2. Recommended pilot loop architecture for separating hydraulic, chemical, and downstream effects [81].

The pilot loop in Figure 11.2 places pressure and temperature measurements immediately around the cavitation device, includes gas conditioning and sampling, and provides a downstream hold or reaction volume. This separation is important because radicals are generated in the cavitation zone but longer lived species can continue reacting downstream. A heat exchanger controls the temperature history so that chemical improvements are not confused with simple warming. A separator or quench point permits mass balance on volatile compounds, solids, or phases.

Figure 11.3. Multistage controlled flow hydrodynamic cavitation device with repeated flow passages and recovery regions. [17].

Multistage controlled flow devices, such as the arrangement shown in Figure 11.3, are intended to create repeated cavitation events while maintaining continuous throughput [17]. Staging can distribute the total pressure drop, provide controlled recovery between elements, and improve exposure uniformity. It can also increase cumulative erosion and pumping demand. The correct number of stages is determined by incremental chemical yield, not by the largest feasible pressure drop.

Figure 11.4. Multiscale workflow for translating sonochemical hydrodynamic cavitation from bench to plant [81].

Table 11.1. Scale up variables that should be preserved or explicitly reoptimized

Category	Variable	Why it matters	Scale up approach
Hydraulic	Pressure ratio and recovery profile	Sets bubble growth and collapse history	Use pressure taps and validated CFD, not pump discharge pressure alone
Hydraulic	Velocity and Reynolds number	Controls turbulence, separation, and nuclei transport	Maintain relevant range or characterize deviation
Bubble population	Nuclei concentration and dissolved gas	Sets active bubble number and cushioning	Condition gas and characterize feed
Thermodynamic	Temperature and vapor pressure	Controls vapor loading and physical properties	Use heat removal and matched temperature history
Chemical	Probe or target concentration	Changes scavenging, partition, and oxidant demand	Scale at matched composition and conversion target
Exposure	Pass number or residence time distribution	Determines cumulative treatment	Use tracer tests and per pass sampling
Energy	Hydraulic and electrical specific energy	Determines economics and comparison	Report both device and whole system boundaries
Materials	Wall stress and erosion rate	Can contaminate product and limit life	Coupons, inspection, replaceable liners
Control	Backpressure and gas holdup	Moves cloud and changes collapse location	Closed loop pressure and gas control
Downstream	Hold time, separation, and quench	Long lived chemistry may continue after device	Preserve reaction and separation sequence

Table 11.2. Minimum measurement package for pilot and industrial evaluation

Measurement	Preferred location	Purpose	Common error if omitted
Static pressure	Immediately upstream, throat where possible, and downstream	Define pressure history and Δp	Using pump rating as device pressure
Flow rate	Calibrated line section	Hydraulic power and pass count	Ignoring gas or slurry effect on meter
Electrical power	Motor drive or power analyzer	Plant relevant energy	Using nameplate power
Temperature	Feed, device outlet, tank	Vapor pressure and thermal control	Attributing thermal reaction to cavitation
Dissolved gas/oxygen	Feed and return	Reproducibility and gas chemistry	Uncontrolled air entrainment
Chemical dosimeter	Feed and immediate outlet	Per pass radical activity	Only measuring after long tank residence
Target and products	Time series and outlet	Conversion, selectivity, mineralization	Using color or parent loss alone
Cavitation structure	Optical window or acoustic sensor	Identify regime and collapse location	Assuming more noise means more chemistry
Erosion and metals	Coupons, filters, product	Materials integrity and contamination	Ignoring wear derived catalytic effects
Residence time distribution	Tracer test across loop or reactor	Scale up and kinetic interpretation	Treating total tank volume as active reactor volume