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Table 5 Kinetic expressions derived for ultrasonic parameters

From: Ultrasonic intensification as a tool for enhanced microbial biofuel yields

S.No.

Parameters

Equations

Definitions

References

1

Acoustic energy density (AED)

\({\rm AED} = \frac{P}{V}\)

\(P = mc_{\rm p}{\left(\frac{{\rm d}T}{{\rm d}t}\right)}^{}_{t=0}\)

P absolute ultrasonic power, V volume of the medium (cm3 L−1), m mass, c p specific heat capacity, (dT/dt) range of temperature change during sonication

[111]

2

Ultrasonic intensity (UI) with the influence of diameter of the probe tip

\({\rm UI} = \frac{4P}{{D^{2} }}\)

P absolute ultrasonic power, D diameter of the probe tip

[111]

3

Ultrasonic intensity

\({\rm UI} = \frac{P}{A}\)

UI ultrasonic intensity, P ultrasonic power, A surface area of the probe

[112]

4

Cell disruption at given acoustic power

\(F_{\rm N} = - {\rm exp}\left[ { - \left( {\frac{t}{\alpha }} \right)^{\beta } } \right]\)

F N cumulative fractions of disrupted cells at given acoustic power, t time of ultrasonication, α and β kinetic constants

[113]

5

Strain rate distribution

\(\varepsilon_{{\rm rr}} (r) = - 2 v_{\rm R} R^{2} r^{ - 3}\)

\(v_{{\rm R}} = {{\rm d}R}/ {{\rm d}t}\, {\dot{Q}}_{\rm a} {\rm O}_2 Z\) \(= (2P_{\rm h}/ {3\rho})^{0.5} (R_{\rm m}^3/ R^3-1)^{0.5}\)

rr strain rate distribution during cavity collapse, v R bubble wall velocity, ρ solvent density, P h external pressure, R m initial radius and R instantaneous radius of imploding cavity

[114]

6

Specific Energy input

\(E_{\rm s} = \frac{P t}{{V\, {\rm TS}_{0} }}\)

E s specific energy, P ultrasonic power, t ultrasonic time, V volume of the sample, TS0 initial concentration of total solids

[115]

7

Actual energy produced by ultrasonication

\(Q_{\rm u} = P \times t\)

Q u energy output, P ultrasonic power, t ultrasonic time

[116]

8

Ultrasound dose

\({\rm UD}_{0} = \frac{P \times t}{V}\)

UD0 ultrasonic dose, P ultrasonic power, t ultrasonic time

[116]

9

Sonochemical effectiveness factor(e us)

\(\varvec{e}_{{{\rm us}}} = f,\eta I,{\mathbb{V}}_{\rm us} ,T\)

\({\mathbb{V}}_{\rm us}=V_{\rm us}/V_{\rm tot}\)

f applied frequency, ηI calorimetrically determined power of the transducer, T average temperature in the reactor, \({\mathbb{V}}_{\rm us}\) dimensionless cavitationally active volume, V us volume of the reactor space affected by sonication, V tot total working volume

[110]

10

Bubble dynamics model

Micro-convection:

\({\text{V}}_{\text{turb}} ({\text{r}},{\text{t}}) = \frac{{{\text{R}}^{2} }}{{{\text{r}}^{2} }}\left( {\frac{\text{d}R}{\text{d}t}} \right)\)

V turb velocity of turbulence, P AW pressure amplitude of acoustic wave, R radius of the bubble, dR/dt bubble wall velocity, V b volume of the bubble, ρ L density of the liquid

[31]

Shock waves:

\(P_{\rm AW}(r, t)=\rho_{\rm L}\frac{R}{r}\left[ 2\left(\frac{{\rm d}R}{{\rm d}t}\right)^2+R \frac{{\rm d}^{2}R}{{\rm d}t^2}\right]\)