Pressure amplitude

These values were ehosen since they were the best compromise between the possibility to provtde significant amplitudes of the cyclical pressure loading and the experimental apparatus limits. Infact, the cyclical pressure amplitude gets lower as the frequency gets higher. 

The pressure amplitudes inside the vessel, the load frequencies and the relevant figures displaying the thermographic results for each scan area are listed in table 2. 

The previous subsection described single-experiment perturbations by J-jumps or P-jumps. By contrast, sound and ultrasound may be used to induce small periodic perturbations of an equilibrium system that are equivalent to periodic pressure and temperature changes. A temperature amplitude 0.002 K and a pressure amplitude 5 P ss 30 mbar are typical in experiments with high-frequency ultrasound. Fignre B2.5.4 illustrates the situation for different rates of chemical relaxation with the angular frequency of the sound wave... 

For airborne sound, the reference pressure is 2 X 10" Pa (29 X psi), which is nominally the human threshold of hearing at 1000 Hz. The corresponding sound pressure level is 0 dB. Conversation is about 50 dB, ana a Jackhammer operator is subject to 100 dB. Extreme levels such as a jet engine at takeoff might produce 140 dB at a distance of 3 m, which is a pressure amplitude oi 200 Pa (29 X 10" psi). These examples demonstrate both the sensitivity and wide dynamic range of the human ear. 

With DLE eombustors, the aim is to burn most of the fuel very lean to avoid the high eombustion temperature zones that produee NOx. So these lean zones that are prone to oseillatory burning are now present from idle to 100% power. Resonanee ean oeeur (usually) within the eombustor. The pressure amplitude at any given resonant frequeney ean rapidly build up and eause failure of the eombustor. The modes of oseillation ean be axial, radial or eireumferential, or all three at the same time. The use of dynamie pressure transdueer in the eombustor seetion, espeeially in the low NOx eombustors ensures that eaeh eombustor ean is burning evenly. This is aehieved by eontrolling the flow in eaeh eombustor ean till the speetrums obtained from eaeh eombustor ean mateh. This teehnique has been used and found to be very effeetive and ensures eombustor stability. 

Systolic pressure, or maximum blood pressure, occurs during left ventricular systole. Diastolic pressure, or minimum blood pressure, occurs during ventricular diastole. The difference between systolic and diastolic pressure is the pulse pressure. While diastolic blood pressure has been historically been used as the most relevant clinical blood pressure phenotype, it has now been clearly established that systolic blood pressure is the more important clinical predictor for cardiovascular morbidity and mortality. More recently, additional attention is focussed on the importance of pulse pressure, i.e. the blood pressure amplitude, as a predictive factor for cardiovascular disease. 

In this mode, acoustic-pressure oscillations are similar to those established in a closed organ pipe. The resulting pressure oscillations then couple with the pressure-sensitive combustion processes to further excite the oscillating pressure and thus produce the high-pressure amplitudes. 

The most important parameter in the analysis of pressure-coupled combustion instability is the acoustic admittance Y, which is the ratio of the amplitude of the acoustic velocity V to the amplitude of the acoustic pressure amplitude of the acoustic velocity V to the amplitude of the acoustic pressure P ... 

Thus, the exponential growth constant of the pressure oscillation is directly related to the acoustic admittance of the propellant. Hence, the acoustic admittance can be evaluated directly from the growth rate of the pressure amplitude. Ryan (R5) has also desired this espression on the basis of acoustic-energy considerations. 

When the pressure amplitude of an acoustic wave in liquid or solid exceeds the ambient pressure (atmospheric pressure), the instantaneous pressure becomes negative during the rarefaction phase of an acoustic wave. Negative pressure is defined as the force acting on the surface of a liquid (or solid) element per surface area to expand the element [3,4]. For example, consider a closed cylinder filled with liquid... 

In Fig. 1.1, the parameter space for transient and stable cavitation bubbles is shown in R0 (ambient bubble radius) - pa (acoustic amplitude) plane [15]. The ambient bubble radius is defined as the bubble radius when an acoustic wave (ultrasound) is absent. The acoustic amplitude is defined as the pressure amplitude of an acoustic wave (ultrasound). Here, transient and stable cavitation bubbles are defined by their shape stability. This is the result of numerical simulations of bubble pulsations. Above the thickest line, bubbles are those of transient cavitation. Below the thickest line, bubbles are those of stable cavitation. Near the left upper side, there is a region for bubbles of high-energy stable cavitation designated by Stable (strong nf0) . In the brackets, the type of acoustic cavitation noise is indicated. The acoustic cavitation noise is defined as acoustic emissions from... 

Fig. 1.2 Numerically simulated frequency spectra of the hydrophone signal due to acoustic cavitation noise. The driving ultrasound is 515 kHz in frequency and 2.6 bar in pressure amplitude, (a) For stable cavitation bubbles of 1.5 pm in ambient radius, (b) For transient cavitation bubbles of 3 pm in ambient radius. Reprinted from Ultrasonics Sonochemistry, vol. 17, K. Yasui, T. Tuziuti, J. Lee, T. Kozuka, A. Towata, and Y. lida, Numerical simulations of acoustic cavitation noise with the temporal fluctuation in the number of bubbles, pp. 460-472, Copyright (2010), with permission from Elsevier...

While the secondary Bjerknes force is always attractive if the ambient radius is the same between bubbles, it can be repulsive if the ambient radius is different [38]. The magnitude as well as the sign of the secondary Bjerknes force is a strong function of the ambient bubble radii of two bubbles, the acoustic pressure amplitude, and the acoustic frequency. It is calculated by (1.5). 

Fig. 1.4 The calculated results for one acoustic cycle when a bubble in water at 3 °C is irradiated by an ultrasonic wave of 52 kHz and 1.52 bar in frequency and pressure amplitude, respectively. The ambient bubble radius is 3.6 pm. (a) The bubble radius, (b) The dissolution rate of OH radicals into the liquid from the interior of the bubble (solid line) and its time integral (dotted line). Reprinted with permission from Yasui K, Tuziuti T, Sivaknmar M, Iida Y (2005) Theoretical study of single-bubble sonochemistry. J Chem Phys 122 224706. Copyright 2005, American Institute of Physics...

Hatanaka et al. [50], Didenko and Suslick [51], and Koda et al. [52] reported the experiment of chemical reactions in a single-bubble system called single-bubble sonochemistry. Didenko and Suslick [51] reported that the amount of OH radicals produced by a single bubble per acoustic cycle was about 10s 106 molecules at 52 kHz and 1.3 1.55 bar in ultrasonic frequency and pressure amplitude, respectively. The result of a numerical simulation shown in Fig. 1.4 [43] is under the condition of the experiment of Didenko and Suslick [51]. The amount of OH... 

In some literature, there is a description that a bubble with linear resonance radius is active in sonoluminescence and sonochemical reactions. However, as already noted, bubble pulsation is intrinsically nonlinear for active bubbles. Thus, the concept of the linear resonance is not applicable to active bubbles (That is only applicable to a linearly pulsating bubble under very weak ultrasound such as 0.1 bar in pressure amplitude). Furthermore, a bubble with the linear resonance radius can be inactive in sonoluminescence and sonochemical reactions [39]. In Fig. 1.8, the calculated expansion ratio (/ max / Rq, where f max is the maximum radius and R0 is the ambient radius of a bubble) is shown as a function of the ambient radius (Ro) for various acoustic amplitudes at 300 kHz [39]. It is seen that the ambient radius for the peak in the expansion ratio decreases as the acoustic pressure amplitude increases. While the linear resonance radius is 11 pm at 300 kHz, the ambient radius for the peak at 3 bar in pressure amplitude is about 0.4 pm. Even at the pressure amplitude of 0.5 bar, it is about 5 pm, which is much smaller than the linear resonance radius. 

In Fig. 1.9, the results of numerical simulations at 300 kHz and 3 bar in ultrasonic frequency and pressure amplitude, respectively are shown as a function of ambient radius [39]. In Fig. 1.9a, the temperature inside a bubble at the end of the bubble collapse is shown with the molar fraction of water vapor inside a bubble. 

Fig. 1.9 The calculated results as a function of ambient radius at 300 kHz and 3 bar in ultrasonic frequency and pressure amplitude, respectively. The horizontal axis is in logarithmic scale, (a) The peak temperature (solid) and the molar fraction of water vapor (dash dotted) inside a bubble at the end of the bubble collapse, (b) The rate of production of oxidants with the logarithmic vertical axis. Reprinted with permission from Yasui K, Tuziuti T, Lee J, Kozuka T, Towata A, Iida Y (2008) The range of ambient radius for an active bubble in sonoluminescence and sonochemical reactions. J Chem Phys 128 184705. Copyright 2008, American Institute of Physics...

In a bath-type sonochemical reactor, a damped standing wave is formed as shown in Fig. 1.13 [1]. Without absorption of ultrasound, a pure standing wave is formed because the intensity of the reflected wave from the liquid surface is equivalent to that of the incident wave at any distance from the transducer. Thus the minimum acoustic-pressure amplitude is completely zero at each pressure node where the incident and reflected waves are exactly cancelled each other. In actual experiments, however, there is absorption of ultrasound especially due to cavitation bubbles. As a result, there appears a traveling wave component because the intensity of the incident wave is higher than that of the reflected wave. Thus, the local minimum value of acoustic pressure amplitude is non-zero as seen in Fig. 1.13. It should be noted that the acoustic-pressure amplitude at the liquid surface (gas-liquid interface) is always zero. In Fig. 1.13, there is the liquid surface... 

When the side wall of a liquid container is thick enough, it can be regarded as a rigid wall. When the side wall is too thin, there is considerable vibration of the side wall caused by an acoustic field in the liquid. Then, the side wall can be regarded as a free surface and the acoustic-pressure amplitude near the side wall becomes nearly zero [86]. 

An ultrasonic horn has a small tip from which high intensity ultrasound is radiated. The acoustic intensity is defined as the energy passing through a unit area normal to the direction of sound propagation per unit time. Its units are watts per square meter (W/m2). It is related to the acoustic pressure amplitude (P) as follows for a plane traveling wave [1]. 

The ultrasound radiated from a horn tip, however, is not a plane wave. The acoustic pressure amplitude is more accurately calculated by Eq. (1.21) along the symmetry axis [1, 89]. 

Fig. 1.16 Calculated radius of a bubble in a bubble cloud as a function of time for one acoustic cycle at 29 kHz and 2.36 bar in frequency and pressure amplitude of ultrasound, respectively. The ambient radius is 5 pm. The dotted curve is the calculated result for an isolated bubble. The dashed one is the calculated result with the interaction only with neighboring bubbles. The solid one is the calculated result taking into account all the interactions with surrounding bubbles. Reprinted figure with permission from Yasui K, Iida Y, Tuziuti T, Kozuka T, Towata A (2008) Strongly interacting bubbles under an ultrasonic horn. Phys Rev E 77 016609 [http /Aink.aps.org/abstract/PRE/v77/ e016609]. Copyright (2008) by the American Physical Society...

Since it is necessary for the negative pressure in the rarefaction cycle to overcome the natural cohesive forces acting in the liquid, any increase in these forces will increase the threshold of cavitation. One method of increasing these forces is to increase the viscosity of the liquid. Tab. 2.1 shows the influence of viscosity on the pressure amplitude (Pft) at which cavitation begins in several liquids at 25 °C, at a hydrostatic pressure of 1 atm. 

What is apparent from Fig. 2.23 is that the fate of the bubble, i. e. whether it remains as a stable bubble or is transformed into a transient, depends upon many factors such as temperature, vapour pressure, hydrostatic pressure, acoustic pressure amplitude... 

In general an increase in intensity (I) will provide for an increase in the sonochemical effects. Cavitation bubbles, initially difficult to create at the higher frequencies (due to the shorter time periods involved in the rarefaction cycles) will now be possible, and since both the collapse time (Eq. 2.27), the temperature (Eq. 2.35) and the pressure (Eq. 2.36) on collapse are dependent on P i(=Ph + PA)> bubble collapse will be more violent. However it must be realised that intensity cannot be increased indefinitely, since (the maximum bubble size) is also dependent upon the pressure amplitude (Eq. 2.38). With increase in the pressure amplitude (P ) the bubble may grow so large on rarefaction (R g, ) that the time available for collapse is insufficient. 

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