Total erosion efficiency (ε) [7]
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He graduated and obtained his Ph.D. in Applied Life Sciences from Tokyo University of Agriculture and Technology (Japan) in 2011. He was awarded Japanese government scholarship and he visited University of California at Davis (UCD) as an exchange student in 2010. After his graduation, he became a research fellow at the German Cancer Research Center (Deutsches Krebsforschungszentrum, DKFZ) in Heidelberg (Germany). Dr. Ying acts as a reviewer of many scientific journals and has authored or co-authored over 25 scientific publications. 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If the dynamic alteration of the absolute static pressure reaches or drops below the vapour pressure of the liquid, vapour bubbles are formed inside the liquid and can collapse as they move to a high-pressure region [1]. When the vapour bubbles collapse, shock waves are produced that propagate at the speed of sound through the liquid [2]. So, cavitation can produce undesirable effects such as noise, vibration, pressure fluctuation, erosion and efficiency loss in a hydraulic system.
The pressure difference due to the dynamic effect of the fluid motion is proportional to the square of the relative velocity. This can be written in the form of the usual pressure coefficient (Cp), presented in the Equation 1 [3]:
where ρ is the density of the liquid; vo and po are the velocity and pressure of an undisturbed liquid, respectively, and (p-po) is the pressure differential due to dynamic effects of fluid motion.
If pressure p reaches a minimum, p=pmin (see Figure 1), the pressure coefficient will be minimum, and a set of conditions can be created so that pmin drops to a value where cavitation can begin. This can be accomplished by raising the relative velocity vo for a fixed value of the pressure po or lowering po with vo remaining constant. If surface tension is ignored, the pressure pmin will be the pressure inside the cavity. This pressure will be the bubble pressure. If we consider that cavitation will occur when the normal stresses at a point in the liquid are reduced to zero, then the bubble pressure will assume the value of the vapour pressure, pv. Then, the cavitation index (σ) is the theoretical value of the negative pressure coefficient, Equation 2:
The cavitation index can designate the probability of a system to cavitate and establish different intensity levels of cavitation in this system.
Pressure distribution in a submerged body [4].
As the pressure is increased, the bubble diameter decreases from the original size reaching a minimum size. The process takes place until the bubble diameter becomes microscopic. If the bubble collapses, then shock waves form with celerity equal to the speed of sound in water. If a boundary is close to where the bubble imploded, it will deform into a microjet. The velocity of this microjet is high and the shockwave produces high pressure responsible for the cavitation damage of a surface [5]. The collapse of the cavities formed by the cavitating jet generates high-pressure waves, estimated to be approximately 69.00 MPa [6] and high-speed microjets (above 100 m/s), all of which have a significant amount of destructive power [7].
Flow conditions leading to the onset of cavitation are generally conservative in predicting damage. The severity of the damage may be related to both intensity of cavitation and exposure time [8].
In hydraulic structures, as the high velocity flow passes over the many irregularities that exist in the concrete surface, cavitation can commence and consequently, damages may occur. Therefore, the material properties of the surface have to be improved to provide adequate resistance either during the construction phase or when the substitution of the eroded concrete is required.
Particularly of interest, in developing countries, hydropower will continue to play a significant role in supplying energy. Therefore, as energy demands are continuously increasing, new hydropower plants (dams and appurtenances) are being constructed. Nevertheless, the operation and maintenance of these structures, which include the spillways and tunnels under high velocity flows, are a great issue. Therefore, the material properties have to be improved to provide adequate resistance, either during the construction phase or substitution phase. For example, the substitution of a great area of eroded concrete was required in the case of Porto Colombia hydropower spillway and dissipation basin in the Grande River, Brazil [7].
An additional complexity is observed in hydraulic structures due to the simultaneous effect of cavitation and high impact of the flow [9,10,4]. Evaluation of the concrete resistance erosion in hydraulic structures is essential to guarantee adequate operation. It was suggested by [11] defining a methodology to appropriately test the materials, when submitted to cavitation, to be used in hydraulic structures.
In this chapter the authors discuss the use of high velocity cavitating jets to determine the erosion in high performance concretes for hydraulic structures. Moreover, two alternatives uses for cavitating jets are presented: 1) the inactivation of Escherichia coli, and, 2) the decomposition of persistent compounds in water.
Evaluation of the concrete erosion resistance in hydraulic structures is essential to guarantee adequate operation, and tests should be performed under the same operation conditions. Various techniques can be used to induce cavitation, such as, ultrasonic methods, hydrodynamic methods, and high-speed/high-pressure homogenization. In the hydrodynamic cavitation, pressure variations are produced using the geometry of the system, while in the acoustic cavitation pressure variations are effected using sound waves.
Concrete resistance to cavitation damage was investigated by [12-14] using a Venturi device. A chamber was used by [15] to test concrete samples subjected to short-duration cavitation. An alternative test for erosion evaluation uses water cavitating jet technology [16,17] to achieve short test time. A flow cavitation chamber was used by [18-20] to test concrete and different rock samples subjected to short-duration cavitation. Water cavitating jet technology was firstly used to clean surfaces. For example, [21] compared the efficiency of conventional non-cavitating jets to cavitating jets.
Regardless several researchers have worked with the cavitation jet apparatus, to test concretes submitted to the erosive effect of cavitation, the appropriate nozzle specifications and sample dimensions must be experimentally determined.
The cavitating jet apparatus uses a nozzle specially designed to produce cavitation, combining high-velocity flows and cavitation (with an appropriate cavitation index). Because the collapse of bubbles is concentrated over a microscopic area, localized stresses are produced, providing the cavitating jet with a great advantage over steady non-cavitating jet operating at the same pressures and flow rates [22]. Figure 2 shows a schematic of the formation of a cavitating jet. As the high velocity flow leaves the orifice nozzle, eddies are formed between the high velocity layer of the jet and the surrounding liquid. If a nucleus in the water is captured in one of these eddies and the pressure inside the eddy drops to vapour pressure, then the nucleus will begin to grow. However, if the pressure remains near the vapour pressure long enough for the nuclei to reach the critical diameter, then it begins to grow almost by vaporization. As long as the size of the vapour cavity increases, the strength of the eddy is destroyed and the rotational velocity decreases. Then, the surrounding bubble pressure is no longer the vapour pressure. Inside the cavity the pressure remains at vapour pressure and cavity surroundings at hydrostatic pressure. Consequently, the cavity becomes unstable and it collapses inward.
Schematic flow of a submerged jet [6].
Now, consider in Figure 3 a spherical vapour bubble of initial radius b and internal pressure Po. Later on, the bubble acquires the radius re at pressure Pe. If a small amount of air is in the liquid, such that the cavities are filled exclusively by vapour, the bubble growth and collapse is intense and causes severe damages to the vicinity [6].
The pressure magnitude generated at the solid boundary is expected to be a main factor in estimating the erosion efficiency of a cavitating jet impinging on a solid boundary as shown in Figure 3.
Cavitating jet formation and bubble collapse (Adapted from [23])
For a steady water jet (vj),the generated pressure magnitude (Ps) can be estimated from the stagnation pressure, as in Equation 3:
A moderate level pressure would be inappropriate for testing erosion rates. Depending on the standoff distance, water droplets formation can occur and the impact pressure can be larger than Ps. It was indicated that the transient pressure (Pa) between the cavity implosion and the solid boundary could be approximated by the “water hammer” equation, as in Equation 4:
with c as the sound velocity in water.
However, the pressure magnitude generated with the impact of a cavitating jet (Pb) is different from that generated by the impact of distinct droplets. According to [21] the maximum pressure Pb develops if a bubble collapses in an incompressible flow, under isothermal conditions, resulting in equation (5):
where α relates the gas pressure inside the bubble in the beginning of collapse under the pressure Ps. Therefore, the cavitating jet erosion efficiency R* is related to Pb and Pa in Equation (6):
Equation (5), however, does not take into account the geometry and finishing of the nozzle. It was verified by [10] that the finish and the geometry of the nozzle influence the erosion performance of the equipment. The author accomplished a series of tests with a cavitating jet apparatus, verifying the influence of these two variables in aluminium and concrete samples. Therefore, in Equation (7) a geometric efficiency factor η was introduced in the cavitation erosion efficiency [7] to calculate the total efficiency η :
To simulate the cavitation phenomenon and to evaluate the erosion in samples, a cavitating jet apparatus was constructed [10] in the Laboratory of Hydraulics and Fluid Mechanics at the State University Campinas, Brazil. Figure 5 and 6 shows a schematic representation and a photo of the test facility. A high-pressure displacement pump (kept at pressure 12.00 MPa) conducts water to the facility and to the metallic chamber provided with two windows for visualization of the tests. A pressure-regulating valve guaranteed safe operation. At the end of the high-pressure pipe and inside of the inactivation chamber, the nozzle with a hole (diameter of 2.00 x 10-3 m) was positioned at the end of the high-pressure pipe.
The nozzle specifications must be established according to its use and they are determined experimentally. The researcher [10] used three nozzle geometries (20° conical, 132° conical and circular) with both shaped and rounded edges (Figure 4).
Nozzle geometries (a) conical 20°, (b) conical 132°, (c) circular [7].
The facility is supplied with no re-circulated water. The samples are placed in the chamber subjected to the cavitating jet, that is, the jet is impinging onto the sample surface. The chamber is filled with water in order to allow the occurrence of the bubbles implosion. The damaged area was measured, the pits number were counted and the samples were photographed every 60 seconds until 300 seconds of time test. Then, the test continued more 900 uninterrupted seconds, achieving a total test time of 1200 seconds.
Apparatus test parameters can be adjusted to obtain optimal efficiency. In this experiment, pressure and velocity of the system were adjusted to reach a cavitation index of 0.14, which is considered to cause damage in hydraulic structures [11]. The three different geometry nozzles were used to evaluate the efficiency of cavitating jets.
Schematic of experimental setup for erosion tests.
Photo of experimental setup in the Laboratory.
Concrete is a heterogeneous material, which poses difficulties to test and evaluate optimal conditions based on comparison test. Thus, aluminum samples were used to establish the nozzle efficiency because the metal is homogenous and allows a comparison among tests in order to evaluate the differing optimal erosion conditions, and damages. Cavitation erosion rates can be analyzed in terms of number of pits by time in order to obtain quantitative and qualitative information on the erosion intensity variation. A binocular magnifier 160x was used to count pits in the surface of the aluminum samples through the tests
First, the apparatus tests were run with the different nozzles presented in Figure 4. The typical value of α=1/4 used by [21] was adopted and maintained for comparison purposes. The sound velocity c in water at 20°C was assumed to be 1482 m/s and water density as 998.2 kg/m3. Given the value of the jet velocity vj, calculations were done for all the data and the efficiencies η were calculated. It was presented by [7] the efficiency results for the nozzles (Table1).
\n\t\t\t\tNozzle\n\t\t\t | \n\t\t\t\n\t\t\t\tSharp-edge (ε)\n\t\t\t | \n\t\t\t\n\t\t\t\tChamfered- edge (ε)\n\t\t\t | \n\t\t
132°Conical | \n\t\t\t0.43 | \n\t\t\t0.17 | \n\t\t
Circular | \n\t\t\t0.33 | \n\t\t\t0.12 | \n\t\t
20°Conical | \n\t\t\t0.16 | \n\t\t\t0.12 | \n\t\t
Total erosion efficiency (ε) [7]
According to Table 1, better results were observed for sharp-edged nozzles than for chamfered-edge ones. The best total efficiency was obtained for the 132° conical nozzle (ε=0.43). Also, the time intervals required to accomplish the same erosion rate were observed to be 300 seconds for 132° conical sharp-edged nozzle, and 1200 seconds for the chamfered-edge nozzle. Therefore, not only the entrance pressure, but also nozzle geometry and finish are important for optimizing the apparatus efficiency. The authors pointed out that obtuse-angle nozzles with sharp-edge are preferable to perform cavitation erosion tests.
Using cavitating jet apparatus (σ=0.14) and the 132° conical sharp-edged nozzle, three special concrete samples were tested to obtain the cavitation erosion rate. The procedure of making samples begins by separating, cleaning and stocking the aggregates in the laboratory the day before. The next day, samples were molded in agreement with the Brazilian Standard NBR 5738. Table 2 presents samples compositions and compressive resistance obtained.
\n\t\t\t\tSample\n\t\t\t | \n\t\t\t\n\t\t\t\tCement and Aggregates\n\t\t\t\t \n\t\t\t\t(Relation)\n\t\t\t | \n\t\t\t\n\t\t\t\tWater and Cement\n\t\t\t\t \n\t\t\t\t(Relation)\n\t\t\t | \n\t\t\t\n\t\t\t\tAggregate\n\t\t\t | \n\t\t\t\n\t\t\t\tAddition\n\t\t\t | \n\t\t\t\n\t\t\t\tCompressive\n\t\t\t\t \n\t\t\t\tResistance (MPa)\n\t\t\t | \n\t\t
A | \n\t\t\t1:4 | \n\t\t\t0.3 | \n\t\t\tLimestone | \n\t\t\t- | \n\t\t\t55.00 | \n\t\t
B | \n\t\t\t1:4 | \n\t\t\t0.3 | \n\t\t\tGranite | \n\t\t\t- | \n\t\t\t63.00 | \n\t\t
C | \n\t\t\t1:4 | \n\t\t\t0.3 | \n\t\t\tGranite | \n\t\t\tSilica * | \n\t\t\t83.00 | \n\t\t
Characteristics of the concrete samples.
* 8% the mass of cement
Figure 7 shows the erosion comparison between concrete samples in terms of volume erosion in time.
Concrete erosion over time.
The best results were obtained for sample C which contains hard aggregate, superior axial compressive resistance and the addition of silica. However sample A (fck 55.00 MPa) presented better results than sample B (fck 63.00 MPa), despite of the greater compressive resistance, showing that an adequate concrete (resistant to cavitation erosion) is a combination of several factors such as aggregate type, size and shape, additions to cement and water cement relation. Figure 8 shows the evolution of erosion in the B concrete sample. Additional tests were performed with unsubmerged and submerged samples, showing that the formation of a high-velocity cavitating jet provoked high erosion rates in the sample as expected, according [7]. In [24] the authors compared the erosion generated by the cavitating jet imping directly in to and parallel to a concrete sample. In this last case, the sample is not subjected to the effects of the impact force of the jet. Comparing the results, a higher erosion rate was observed when the samples are positioned directly over to the cavitating jet (Figure 9).
Erosion evolution of concrete sample B over time.
Concrete sample placed parallel to the cavitating jet [24].
Fast, efficient and economic testing is needed. The use of the cavitating jet allowed significant reduction in testing times, [10] specially when compared to the Venturi device. In [12] the author took 30 hours to compare the cavitating wear among concrete samples. As concrete is a heterogeneous material, samples are necessary to be larger than a few millimeters, as used by [25] to test metal samples. [10] used samples 20 cm in diameter and 5 cm in height to test concrete samples. The cavitating jet apparatus used in this experiment is compact, low cost and has short test times. It can also be used to measure erosion for testing concrete use in hydraulic structures. The cavitating jet apparatus creates a force larger than the one generated by a simple jet of high pressure, and thus can simulate the combined effect of high-speed flows and cavitation normally experienced in hydraulic structures
Water quality has deteriorated over the years due to industrial and agricultural pollution as well as an increase in the domestic sewage generated by a rapidly growing population. Pathogenic microorganisms in water pose a serious health and security threat to drinking water supply systems. The quality of water for human consumption can be improved by controlling pollution and by increasing the efficiency of inactivation techniques, which involves the destruction of pathogens present in water at a reasonable cost.
The use of chemicals, such as chlorine gas, sodium hypochlorite, calcium hypochlorite, hydrated ammonia, ammonium hydroxide, ammonium sulphate, and ozone, is common during the water disinfection process in conventional water treatment systems [26]. Many problems arise when using chemical methods for disinfection, including the high maintenance demands of the associated facilities (corrosion, incrustation), the formation of toxic byproducts (chlorine addition may generate byproducts such as trihalomethanes) and environmental concerns (chemical effluents released into rivers compromise aquatic life) [27]. Therefore, the development of alternative techniques for inactivating pathogens in water is desirable. Inactivation based on the cavitation phenomenon appears to be a promising alternative or supplement to existing techniques [28].
Research using the phenomenon of cavitation to inactivate microorganisms has been performed in the United States, Russia, India, China, Japan, UK, South Africa, France, Mexico, among others, using different methods. The hydrodynamic cavitation method is the most studied and disseminated. This is because the best results are achieved. However, there is currently no standardized method to carry out inactivation with this technique with respect to the pressures and speeds testing.
The performance shown by the studies on the wear of concrete indicated that the cavitating jet apparatus could be adapted to test the inactivation of microorganisms and the decomposition of persistent compounds in water. However, tests are necessary to establish appropriate pressures and velocities for each study. The inactivation of microorganisms was performed with Escherichia coli that are microorganisms commonly found in the intestines of humans and other warm-blooded animals and indicate the presence of water contamination. The decomposition of persistent compounds was performed with methylene blue.
As illustrated in Figure 2 the cavitating jet apparatus with 132° conical nozzle was adapted to conduct the experiments to inactivate Escherichia coli and to decompound the persistent methylene blue compound. The chamber used was 700.00 x 10-3 m high with a diameter of 300.00 x 10-3 m. Part of the container was filled with 40.00 x 10-3 m3 of contaminated water to be studied in a closed circuit. A low-cost refrigeration system was adapted to the apparatus. A 12.70 x 10-3 m diameter and 19,500.00 x 10-3 m long copper coil was placed inside of the inactivation chamber. This coil was connected to a reservoir with a capacity of 750.00 x 10-3 m3 that was filled with clean water. The temperature of the tests was held constant at 33 °C, thus assuring that the inactivation was exclusively due to the cavitation phenomenon instead of an increase in the water temperature.
All of the microbiological, physical, and chemical procedures of this study follow the procedures described in the Standard Methods for Examination of Water and Wastewater Manual [19,20]. Non-pathogenic Escherichia coli (ATCC 25922) were used for the microbiological tests. Viable cell counting was performed using the Colilert Method®. The presented results of the microbiological tests are the average of the results obtained in the dilutions.
The high-pressure displacement pump recirculated the water to be treated and was kept at a pressure of 4.00, 8.00, 10.00 and 12.00 MPa. First, the inactivation device was turned on without pressure to circulate the residue being treated. Samples were then collected at the initial time (T0) to obtain a control sample. Next, a vial of the sample was removed to measure the number of viable Escherichia coli cells. In addition to the control time point (T0), samples were collected every 900 seconds. All of the tests were repeated in triplicate, however, the results represent an average of the individual results.
Figure 10 shows that the higher the pressure, the higher the inactivation. At time test 1800 seconds, the following inactivation rates are observed: less than 37.50% for pressure test 4.00 MPa, 98.30% for test pressure 8.00 MPa, 99.96% for pressure test 10.00 MPa, and 100% for pressure 12.00 MPa,. Although the inactivation percentages are close to pressures 8.00, 10.00 and 12.00 MPa, the highest inactivation rate was achieved for test pressure 12.00 MPa.
Inactivation rate and pressure.
However, the optimum inactivation rate will be achieved by relating the inactivation rate to the energy consumption. The Energy Efficiency (EE) of the apparatus is calculated with Equation 8 and the experiment results are shown in Table 3.
being:
Cf: Final Concentration of Escherichia coli (CFU/mL);
Ci: Initial Concentration of Escherichia coli (CFU/mL);
∀: Volume (mL);
P: Power (W);
T: Time (seconds).
\n\t\t\t\tParameters\n\t\t\t | \n\t\t\t\n\t\t\t\tTest pressures\n\t\t\t | \n\t\t|||
\n\t\t\t\t4.00 MPa\n\t\t\t | \n\t\t\t\n\t\t\t\t8.00 MPa\n\t\t\t | \n\t\t\t\n\t\t\t\t10.00 MPa\n\t\t\t | \n\t\t\t\n\t\t\t\t12.00 MPa\n\t\t\t | \n\t\t|
t (s) | \n\t\t\t1800.00 | \n\t\t\t1800.00 | \n\t\t\t1800.00 | \n\t\t\t1800.00 | \n\t\t
V (mL) | \n\t\t\t40000.00 | \n\t\t\t40000.00 | \n\t\t\t40000.00 | \n\t\t\t40000.00 | \n\t\t
i (A) | \n\t\t\t25.00 | \n\t\t\t29.00 | \n\t\t\t31.00 | \n\t\t\t32.80 | \n\t\t
U (V) | \n\t\t\t220.00 | \n\t\t\t220.00 | \n\t\t\t220.00 | \n\t\t\t220.00 | \n\t\t
P (W) | \n\t\t\t5500.00 | \n\t\t\t6380.00 | \n\t\t\t6820.00 | \n\t\t\t7216.00 | \n\t\t
Ci (CFU/mL) | \n\t\t\t918425.00 | \n\t\t\t141360.00 | \n\t\t\t800013.00 | \n\t\t\t256547.00 | \n\t\t
Cf (CFU /mL) | \n\t\t\t574000.00 | \n\t\t\t2410.00 | \n\t\t\t337.00 | \n\t\t\t9.00 | \n\t\t
Ci-Cf (CFU /mL) | \n\t\t\t344425.00 | \n\t\t\t138950.00 | \n\t\t\t799676.00 | \n\t\t\t256538.00 | \n\t\t
(Ci-Cf)/P(CFU /mL)/(W) | \n\t\t\t62.62 | \n\t\t\t21.78 | \n\t\t\t117.25 | \n\t\t\t35.55 | \n\t\t
ЄЄ (CFU /J) | \n\t\t\t1391.62 | \n\t\t\t483.98 | \n\t\t\t2605.66 | \n\t\t\t790.03 | \n\t\t
Energy Efficiency to inactivate Escherichia Coli at different pressure.
In order to compare the tests, a time test of 1800 seconds was standardized. Table 3 indicates that for the conical 132° nozzle, the most efficient operation pressure is 10.00 MPa, followed by pressure 4.00, 12.00 and 8.00 MPa.
A complementary study was conducted with natural waters retrieved from a lake at the University’s campus [28]. Given past studies by [29,30], high efficiency for the first 900 seconds of the test with an inactivation rate of 90% was obtained.
The substance methylene blue was used for testing degradation of persistent compounds. The results for each time and pressure test were analyzed using the spectrophotometer (Hach DR / 4000U Spectrophotometer) scan from 800 nm to 200 nm. To analyze the degradation of persistent compounds it is necessary to observe the decrease in absorbance at a given wavelength band over time. The absorbance is proportional to the concentration of the substance based on the law of Lambert-Bee. Changes in the molecule of methylene blue are then analyzed for two wavelength bands: the ranges from 200nm to 400nm, ultraviolet (UV), and the range from 400nm to 800nm, visible to the human eye.
The result of the degradation is shown in Figures 11 and 12 at a pressure of 12.00 MPa and time 1800 seconds. Figure 11 shows no significant reduction in absorbance. Figure 12 indicates a reduction of 10.5% in the final absorbance at the peak wavelength range of 664nm.
Decomposition of methylene blue, pressure 12 MPa, UV wavelength.
Decomposition of methylene blue, pressure 12 MPa, visible wavelength.
Comparing the results of Figures 11 and 12, the greatest difference in absorbance occurs in the visible band. Pressures lower than 12.00 MPa did not show any significant decrease in the absorbance. At the end of the tests, the residue was placed in translucent plastic bottles and exposed to sunlight. It was observed that the methylene blue dye was then degraded only by the peroxidation generated by the cavitation treatment.
The synergy between the cavitation process and the oxidative processes is a current and relevant research [31-33]. Recent work by [34-36] has been focusing on the degradation of persistent pollutants, such as pharmaceuticals, dyes, industrial effluents, pesticides, in addition to research with microorganisms.
Cavitation is known for its undesired effects produced in hydraulic systems, namely, the noise, vibrations, pressure fluctuations, erosion and efficiency loss. In hydraulic structures, cavitation can be destructive. Many times, the materials to do repairs in the structure are used without the necessary laboratory tests. These tests can be onerous, showing the importance of previous studies to check the applicability, either for the project phase or for repairs.
Despite the undesirable effects of cavitation, this phenomenon showed to be an interesting alternative to inactivate bacteria in water. The use of chemicals, such as chlorine, is commonly used for water disinfection in conventional water treatment systems. Cavitation can be used as a “clean” treatment, as it can reduce the quantity of chemicals added to water in the conventional water treatments.
The cavitating jet equipment simulates cavitation and its efficiency can be improved by using different nozzle configuration that forms the cavitating jet. This work showed that the most efficient nozzle geometry was obtained using a sharp-edge 132° conical nozzle for different applications, erosion tests, bacteria inactivation and decomposition of compounds.
The cavitating jet apparatus proved to be efficient, low cost, low power consumption when used to test material erosion and to inactivate Escherichia coli. Furthermore, it can be adapted to operate at larger scale treatments.
The authors would like to acknowledge FAPESP (São Paulo Research Foundation) for the Scientific Research Scholarship (Process: 2000/03611-0, 2000/03732-2, 2002/10348-0, 2009/53553-1 and 2011/16347-4) and Research Award Scheme (Process: 2009/54278-4).
The interior noise level in a jet aircraft is mainly depend on noise which generated by turbulent boundary layers (TBL), if the rest of noise sources such as ventilation systems, fans, hydraulic systems, etc. have been appropriately acoustically treated. When the aircraft passes through the atmosphere, the turbulent boundary layer creates pressure fluctuations on the fuselage. These pressure fluctuations cause the aircraft fuselage to vibrate. The noise generated by the vibration is then transmitted to the cabin.
\nThe noise emitted by the aircraft fuselage depends on the speed of the vibrating plate, which in turn depends on the speed of the aircraft, the geometry and size of the plates, and the loss or damping of the plates. It is obvious that the acoustic performances of the internal system, trim panels etc., will also affect the noise inside the aircraft. Graham [1] came up with a model in aircraft plates to predict TBL induced noise, in which the modal excitation terms were calculated by an analytical expression. In Graham’s another research [2], the advantages of various models describing the cross power spectral density induced by a flow or TBL across a structure was discussed. Han et al. [3] tried to use energy flow analysis to predict the noise induced by TBL. The method can better predict the response caused by the TBL excitation. However, the noise radiation caused by the flat panel cannot be predicted well. To avoid this deficiency, Liu et al. [4, 5, 6] described a model to predict TBL induced noise for aircraft plates. In their work, the modal excitation terms and acoustical radiation efficiency can be predicted properly and the predicted results are also compared with that of the wind tunnel and in-flight test. Rocha and Palumbo [7] further investigated the sensitivity of sound power radiated by aircraft panels to TBL parameters, and discussed the findings by Liu [4] that ring stiffeners may increase TBL induced noise radiation significantly.
\nThe radiation efficiency of a plate plays an important role in vibro-acoustic problems. In recent related research, the sound medium around the fuselage of the aircraft is often considered to be stationary. Under this assumption, Cremer and Heckl [8] used a more concise formula to predict the acoustic radiation efficiency of an infinite plate. Wallace [9] derived an integral formula based on far-field acoustic radiation power to calculate the modal acoustic radiation efficiency of a finite plate. Kou et al. [10] proposed modifications to the classical formulas given by Cremer and Leppington, regarding the influence of structural damping on the radiation efficiency.
\nA comparison of the acoustic radiation of the plate with stationary fluid and convective fluid-loaded can be found in [11, 12, 13]. Graham [11] and Frampton [12] studied the influence of the mean flow on the modal radiation efficiency of a rectangular plate. They found that at high speeds, as the modal wave moves upstream, the increase of flow velocity would reduce the modal critical frequency. As a consequence, the acoustics radiation efficiency under the critical frequency of the plate would be higher. Kou et al. [13] also conducted a research for the effect of convection velocity in the TBL on the radiation efficiency. Kou et al. found that the modal averaged radiation efficiency will increase with the increase of the convection velocity below the hydrodynamic coincidence frequency. The study also showed that the increase of the structural loss factor could increase the modal average radiation efficiency at the subcritical frequencies, and the damping effect increases with the increase of the flow velocity.
\nFor a plate subjected to a TBL fluctuation, although a large amount of research work used experimental and computational methods for the vibro-acoustical properties of plates, it is worth a chapter to introduce the prediction model and summarize recent findings for TBL induced plate vibrations and noise radiations. The following sections begins with a description of models for the wavenumber-frequency spectrum of TBL, and then a specific presentation of the calculation of vibro-acoustic responses of the wall plate excited by TBL is followed. In the end, the effect of flow velocity (Mc) and structural damping on the modal averaged radiation efficiency is discussed.
\nAs for the research about wavenumber-frequency spectrum of turbulent boundary layer, Corcos [14], Efimtsov [15], Smolyakov-Tkachenko [16], Williams [17], Chase [18, 19] and other researchers put up with a series of widely used of wavenumber-frequency spectrum model. The models are established according to a large number of experimental data and statistical theory of turbulence. The following parts introduce some typical wavenumber-frequency spectrum models.
\nThe model proposed by Corcos during the last few decades has been widely used for many different types of problems. The model is applicable in the immediate neighborhood of the so-called convective ridge [20], as long as ωδ/U∞ > 1. In this expression δ is the thickness of the boundary layer and U∞ the velocity of the flow well away from the structure. The flat-plate boundary layer is taken to lie in the x-y plane of a Cartesian coordinate system, with mean flow in the direction of the x-axis. Corcos assumes that the cross power spectral density, between the pressures at two different positions separated by the vector n can be expressed as
\nwhere Φpp(ω) is the auto-power spectral density of turbulent boundary layer fluctuating pressure, kc = ω/Uc is the convection wave number. γ1 and γ3 can be obtained by fitting experimental data, γ1 and γ3 are 0.11–0.12 and 0.7–0.12 respectively for smooth rigid siding.
\nThe Fourier Transform of ξx and ξy can obtain wavenumber-frequency spectrum
\nSo, the normalized wavenumber-frequency spectrum in wavenumber domain is
\nCaiazzo and Desmet [21] proposed a generalized model which based on the Corcos model. The model uses butterworth filter to replace exponential decay of x and y direction in the Corcos model. It can make the wavenumber-frequency spectrum attenuation rapidly near the convection wave number by adjusting the parameters. Expression of this model is as follows
\nwhere θp = (π/2P)·(1 + 2p), θq = (π/2Q)·(1 + 2q). When P = Q = 1, Eq. (4) is equal to the Corcos model.
\nAnalogously, the normalized wavenumber-frequency spectrum in wavenumber domain is
\nThe Efimtsov model assumes, as in the Corcos model, that the lateral and the longitudinal effects of the TBL can be separated. However, in the Efimtsov model the dependence of spatial correlation on boundary layer thickness, δ, as well as spatial separation is taken into account. Correlation length 1/γ1kc and 1/γ3kc in Corcos model are replaced with Λx and Λy. The Efimtsov model gives the cross power spectral density of the pressure at two different positions separated by the vector ξ as
\nwhere
\nwhere Sh is the Strouhal number and equal to Sh = ωδ/Uτ and Uτ the friction velocity which varies with the Reynolds number but is typically of the order 0.03 U∞–0.04 U∞. At high frequencies these expressions correspond to a Corcos model with γ1 = 0.1 and γ3 = 0.77. Coefficient a1–a7 are 0.1, 72.8, 1.54, 0.77, 548, 13.5 and 5.66 respectively. When 0.75 < M∞ < 0.9, the Λy can be determined by numerical interpolation. At high frequency, the Efimtsov model and the Corcos model are equal while γ1 = 0.10 and γ3 = 0.77.
\nThe normalized wavenumber-frequency spectrum is
\nLike Efimtsov model, Smolyakov-Tkachenko model also takes the boundary layer thickness and scale space separation of boundary layer effect of fluctuating pressure into account. Based on the experimental results, the difference is that the Smolyakov-Tkachenko model amend the space scale function index \n
The normalized wavenumber-frequency spectrum is
\nwhere
\nwhere δ* is the thickness of boundary layer, which is also set as δ* = δ/8.
\nFfowcs-Williams using Lighthill acoustic analogy theory to deduce a frequency-wave spectrum model, in which the speed of the pneumatic equation is set as the source term by Corcos form. A number of parameters in the model and function need further experiments to determine, which is not widely used at present. Hwang and Geib [22] ignore compression factor of the influence of this model to put forward a simplified model. The normalized wavenumber-frequency spectrum is
\nChase’s model is another model commonly used and believed to describe the low-wavenumber domain better than Corcos’s model, which has the same starting point with the Ffowcs-Williams model. The normalized wavenumber-frequency spectrum can be described as
\nwhere
\nFigure 1 shows the comparison of the above models. In the figure, the parameters used by the Corcos model are γ1 = 0.116, γ3 = 0.77, the order of Generalized Corcos model is (P = 1, Q = 4). From the comparison among those models, it can be seen that the Generalized Corcos model attenuates quickly in the vicinity of the convective wave number, and its order is adjustable, which can effectively control the computational accuracy. The model can obtain more accurate prediction results by adjusting parameters. In addition, the Chase model is considered to be able to better describe the pressure characteristics of TBL pulsation at low wave number segment, while other models have some defects at low wave number segment. However, Corcos model is the most commonly used in practical application. Because the model is simple in form and has clear physical significance, a simple calculation formula can usually be obtained when solving the structural vibration and sound response induced by turbulent boundary layer. It should be noted that the structure radiated sound predicted by Corcos model tends to be larger at low wave number.
\nA comparison of models for different wavenumber-frequency spectrum of turbulent boundary layer fluctuating pressure, reproduced from Ref. [23].
Consider a simply supported thin rectangular plate excited by TBL, as shown in Figure 2. In the figure, Uc is turbulent flow velocity, and the direction of the incoming flow is parallel to the X-axis. In this chapter, vibro-acoustic responses are solved by modal superposition method [23].
\nSchematic diagram of simply supported thin rectangular plate excited by TBL.
Assume that point s on the plate is excited by a normal force F at point\n
where \n
The impulse response H satisfies the following governing equation
\nThe impulse response can be expanded as
\nThe modal amplitude of impulse response by using the Galerkin method can be described as
\nCross spectral density of displacement response for any two points on the plate can be defined as
\nwhere
\nIn the above equation, Jmn(ω) is called modal excitation term.
\nWhen using the Corcos model, the coordinate transformation of the quadruple integral in the modal excitation term can be obtained
\nWhere
\nWhen r1 = r2, the auto-spectral density of displacement response can be obtained as
\nAs for vibration (V = jωW) the auto-spectral density is
\nSo, vibration energy and acoustic radiation energy can be expressed as
\nAccording to the definition, the modal average acoustic radiation efficiency excited by TBL of the thin plate is
\nAnother approach to obtain the cross spectral density of vibration response is to solve it directly by using the separable integral property of some turbulent boundary layer pulsating pressure models in the wavenumber domain [24].
\nThe wavenumber-frequency spectrum of TBL satisfies the following relationship
\nwhere \n
The formula can be obtained by substituting the cross spectral density of the vibration response
\nwhere
\nSimilarly, the spectral density of the vibration velocity can be obtained as
\nAs for the Corcos model, we can obtain that
\nwhere
\nAccording to the residue theorem, Λm(ω) and Γn(ω) can be further simplified as
\nVibration energy and sound radiation energy are
\nCompare the above two equations, it can be seen that
\nFinally, the modal average acoustic radiation efficiency can be obtained as
\nBy observing the above equation, it can be found that only the modal excitation term in the modal averaged radiation efficiency is related to turbulence.
\nFigure 3 shows the comparison of two methods for calculating the modal averaged radiation efficiency excited by TBL. The size of the plate is 1.25 × 1.1 m, and the thickness is 4 mm, structural loss factor of aluminum plate is 1%, mach number is 0.5. Obviously, the accuracy of the two methods is equal. Computation speed of analytical method is much faster than integral method, but its range of application has limitations. Only the Corcos model and Efimtsov model can be used to separate integrals in the wave number domain.
\nComparison of calculation methods of the modal averaged radiation efficiency excited by TBL. Reproduced from Ref. [23].
The comparison of measured and predicted velocity spectral density and the radiated sound intensity of a plate (a × b = 0.62 × 0.3 m, and the thickness is 1.1 mm) is shown in Figure 4, which is only compared in narrow band. In this study, the loss factor of the plate assumes as 1.5%. The measured and predicted results for radiated sound intensity and auto spectrum of velocity have a good agreement with the frequency ranges from 100 to 3500 Hz. The agreement of the two type curves provides solid verification to test measured and predicted results.
\nMeasured and predicted velocity auto spectrum and the radiated sound intensity of the plate with the size of a × b = 0.62 × 0.3 m. Narrow band analysis in per Hz. Flow speed 86 m/s.
When the velocity of bending wave in the wall plate is close to the sound velocity in the air, the sound radiation efficiency reaches the maximum value. The corresponding frequency is the so-called critical frequency, and its expression is
\nIn the case of flow, when the velocity of flexural wave propagation in the wall plate is close to the turbulent convection velocity, the wall plate is most excited by the fluctuating pressure of TBL. The corresponding frequency is defined as the hydrodynamic coincidence frequency
\nSimilarly, for order (m, n) mode, its critical frequency and hydrodynamic coincidence frequency are
\nIn conclusion, the relationship between critical frequency and hydrodynamic coincidence frequency can be summarized as follows
\nIn the above two equations, Mc = Uc/c0 is mach number. Subsonic turbulence is generally considered, so the hydrodynamic coincidence frequency is always less than the critical frequency of the plate. It is important to note that the characteristics of frequency is a reference value which is based on the infinite plate hypothesis. Actually, the characteristics frequency of the limited plate slightly higher than a reference value. In addition, for the transverse flow problem, modal power line frequency can be thought of only related to the transverse mode. That is to say, fh,mn ≈ Uckm/2π, where km = mπ/a is lateral modal wave number.
\nThe specific parameters and dimensions used in the calculation are listed in Table 1.
\nPlate length | \na | \n1.25 m | \n
Plate width | \nb | \n1.1 m | \n
Plate thickness | \nh | \n0.002 m | \n
Plate surface density | \nms | \n5.4 kg/m2 | \n
Plate bending stiffness | \nD | \n52 Nm | \n
Air density | \nρ0 | \n1.21 kg/m3 | \n
Sound speed | \nc0 | \n340 m/s | \n
Parameters used in calculation.
The increment of vibration power and acoustic radiation energy are different with the increase of the velocity, which indicates that the changing of velocity can affect the modal averaged radiation efficiency. The modal averaged radiation efficiency of the aluminum plate at three flow velocities (Mc = 0.5; 0.7; 0.9) is shown in Figure 5. It can be seen that when the Mc increases from 0.5 to 0.9, the modal averaged radiation efficiency will increase by 3–7 dB below the hydrodynamic coincidence frequency. And the corresponding hydrodynamic coincidence frequencies (fh) are 1482, 2905, and 4802 Hz, respectively. The results show that the modal averaged radiation efficiency increases in the frequency range below the hydrodynamic coincidence frequency. The increase of the modal averaged radiation efficiency indicates that with the increase of flow velocity, the increment of the radiated sound power is larger than that of the mean square velocity.
\nEffect of the convective Mach number on the modal averaged radiation efficiency of the finite aluminum plate. Reproduced from Ref. [23].
The phenomenon that the modal averaged radiation efficiency increases with the flow velocity can be explained by the hydrodynamic coincidence effect. For the lateral incoming flow problem, the hydrodynamic coincidence is mainly determined by the lateral modal trace speed and the convection velocity. When the bending wave velocity of the lateral mode is the same as the turbulent flow velocity (Uc = 2πf/km), the corresponding hydrodynamic coincidence frequency is f = mUc/2a. Thus a higher convection velocity at the same frequency will lead the TBL excitation to coincide with a lower order lateral mode.
\nThe reason for above phenomenon may be further explored through the modal excitation terms. As illustrated in Figure 6, the lateral modal excitation term (10log10Λm(ω)) is plotted with the lateral mode number (m) and frequency for different flow velocity (Mc). In the figure, the peak of the lateral mode excitation term corresponds to the maximum excitation and its position depends on the hydrodynamic coincidence frequency. The black bold lines in the two sub graphs are the positions where the hydrodynamic coincidence occurs. It can be seen that the slope of the hydrodynamic coincidence line is inversely proportional to the flow velocity, and the higher the velocity is, the lower the order of a certain frequency is. In addition, the lateral modes near the hydrodynamic coincidence line are all strongly excited. As the frequency increases, the number of these modes increases, but the amplitude of their corresponding mode excitation term decreases. Below the critical frequency, a lower order lateral mode always has higher modal averaged radiation efficiency than that of a higher order lateral mode with the same n, since the modal critical frequency moves to lower frequency. So plate with higher flow velocity is supposed to have higher modal averaged radiation efficiency.
\nVariation of the lateral modal excitation term with the lateral mode number and the frequency of a finite aluminum plate. (a) Convective Mc = 0.5 and (b) convective Mc = 0.9. Reproduced from Ref. [13].
As an example, the hydrodynamic coincidence lines for different flow velocity (Mc) and the modal radiation efficiencies of mode (m, 1) are illustrated in Figure 7. The black solid lines in the figure are the hydrodynamic coincidence line corresponding to the mode order and frequency. It can be seen that at a certain frequency, the modal averaged radiation efficiency of the hydrodynamic coincidence modes at higher velocity is always greater than that of the low velocity. In a word, an increase of the flow velocity will increase the modal radiation efficiency of the coincided mode, and then results in the increase of the modal averaged radiation efficiency. Besides, owing to the low pass property of the modal excitation term, the increase of the modal radiation efficiency is restrained above the hydrodynamic coincidence frequency. As a consequence, the modal averaged radiation efficiency is great affected by the flow velocity which only occurs below the hydrodynamic coincidence frequency.
\nHydrodynamic coincidence lines and variation of the modal radiation efficiency with the lateral mode number and the frequency of a finite aluminum plate. m varies, n = 1. Reproduced from Ref. [13].
The modal averaged radiation efficiency changes with structural loss factors for different flow velocity (Mc), as shown in Figure 8. The reference value is calculated according to Leppington’s formula [25]. Though Leppington’s formula is widely used in statistical energy analysis, it does not take the flow and structural damping into account. Figure 8 indicates that an increase of the structural loss factor will increase the modal averaged radiation efficiency under the critical frequency, but the increments are different for different flow velocity. It is found that the modal averaged radiation efficiency is not sensitive to the change of structure loss factor at low Mach number. For example, for a typical high-speed train (Mc = 0.25), the increased modal averaged radiation efficiency is less than 2 dB in the frequency band below the critical frequency when the structural loss factor increases from 1 to 4%. In the case of high flow velocity, the effect of structure loss factor on the modal averaged radiation efficiency is much obvious. When Mc = 0.7, the modal averaged radiation efficiency will increase by about 5 dB if the structural loss factor has the same increment. The results show that the influence of structural damping on the modal averaged radiation efficiency is related to the flow velocity, and the influence of structural damping can be enhanced by increasing the flow velocity.
\nEffect of the structural loss factor on the modal averaged radiation efficiency of a finite aluminum plate. (a) Convective Mc = 0.25 and (b) convective Mc = 0.7. Reproduced from Ref. [13].
The effect of structural damping on the modal averaged radiation efficiency can be qualitatively explained by Eq. (61)
\nEq. (61) shows that the modal averaged radiation efficiency is equivalent to the weighted average function of the modal velocity response, and the weighted coefficient is the modal averaged radiation efficiency. In the frequency band below the critical frequency, the radiation efficiency of each mode varies in the range from 0 to 1. Due to this weighted effect of Eq. (61), the vibration energy (denominator in the equation) decreases more effectively than the acoustic radiation power (molecule in the equation). Thus the radiation efficiency increases in the frequency band below the critical frequency. However, the phenomenon that the radiation efficiency of a damped plate is enlarged with increment of flow velocity has not yet been clearly interpreted.
\nMoreover, it is observed that the effect of structural damping on modal averaged radiation efficiency has a good agreement with the research of Kou [23] at low flow velocity. In their work, it is shown that the modal averaged radiation efficiency of heavily damped structures is sensitive to the change of structural loss factor without turbulent flow. It also implies that Leppington’s equation is not applicable to the prediction of modal averaged radiation efficiency of damped structures at high flow velocity.
\nThis chapter studies the vibro-acoustic characteristics of the wall plate structure excited by turbulent boundary layer (TBL). Based on the modal expansion and Corcos model, the formulas for calculating the modal averaged radiation efficiency are derived. The results indicate that an increment of flow rate will increase the vibration energy and the radiated sound energy of the structure. However, the amplitude of two cases varies with the velocity are not the same, and when the velocity increases, the acoustic radiation efficiency will increase below the hydrodynamic coincidence frequency range. The main reason for this phenomenon is that a higher convection velocity will coincide with lower order modes which have higher radiation efficiencies.
\nThe modal averaged radiation efficiency increases with the increase of structural damping below the critical frequency band. The larger the flow rate, the more significant the effect of structural damping on acoustic radiation efficiency. In the case of low flow velocity, the modal averaged radiation efficiency is not sensitive to the change of structural damping. The structural damping increases from 1 to 4%, and the increase of modal averaged radiation efficiency less than 2 dB. In the case of high flow rate, the modal averaged radiation efficiency will increase by 5 dB when the increment of the structural damping is from 1 to 4%.
\nThanks to the financial support by the Taishan Scholar Program of Shandong (no. ts201712054).
\nFigures 6–8 in this chapter are reproduced from an AIP Publishing journal paper written by the second and third authors, and all the figures are cited in this text.
According to AIP webpage for Copyright and Permission to Reuse AIP materials, AIP Publishing allows authors to retain their copyrights (
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