USE OF RESONANCE AND ANTI RESONANCE FREQUENCY SHIFTS TO LOCATE AND SIZE HOLES IN PIPE AND DUCT WALLS MHF de Salis VIPAC Engineers & Scientists Pty Ltd Unit E1-B, Centrecourt, 5 Paul Street North, North Ryde, NSW 113, Australia. E-mail: maxds@vipac.com.au 1. INTRODUCTION This paper describes a blockage detection previously developed by De Salis and Oldham 1-3 and its successful application to the location and sizing of holes in the walls of circular ducts. The method requires closed-open duct termination conditions with a driver located at the closed end.. RECONSTRUCTION OF THE INTERNAL AREA FUNCTION TO LOCATE A HOLE IN THE DUCT WALL De Salis and Oldham 1-3 demonstrated that the internal profile or blockage area function of a duct can be determined by utilising a single Maximum Length Sequence (MLS) measurement of the loudspeaker voltage to sound pressure transfer function within the duct to reveal the duct resonance and anti-resonance frequencies of the duct. The expression for the blockage area function A b (x)/a 0 (x) of a duct with closed-open end conditions in de Salis and Oldham's work is given by A b(x)/a0(x) Le = [ 1 exp( nπ n=1 nπ x µ (a) n cos L e - n=1 Le χ n cos π ( n-1) ( n 1) π x a 0 L e ) ] (1) where A b (x) is the blockage cross-sectional area, A 0 (x) is the cross-sectional area of the unblocked duct, L e is the end corrected duct length for the closed-open duct and a 0 is an added DC component which is equal to the ratio of blockage to duct volume. As detailed in Reference 1, the value χ n is the n th blockage induced duct resonance value shift and is calculated from the measured resonance frequencies f (b) n of a partially blocked duct and the calculated resonance frequencies f (cu) k of a theoretical unblocked duct of uniform cross-section, identical length and approximately identical volume where
(b) (cu) [ f ] [ f ] ), n k. 4 χ n = ( n k = () c (cu) The unblocked duct resonance frequencies f k are determined from the measured values of f (b) n as follows: N (cu) (k 1) (b) N, fk = fn, k = 1,,3,.. (3) N n= 1 where N is the number of longitudinal resonance frequencies inclusive from N=1 used in the calculation. Similarly, µ ( a) n is the n th blockage induced anti-resonance value shift of the duct and is calculated from the measured anti-resonance frequencies in the partially blocked duct f (b) (a)n and the calculated anti-resonance frequencies f (cu) (a)k of a theoretical unblocked duct of identical length and approximately identical volume but of uniform cross-section, (b) (cu) [ f ] [ f ] ), n k. 4 µ ( a ) n = ( (a)n (a)k = (4) c (cu) The unblocked duct anti-resonance frequencies f (a)k are determined from the measured values of f (b) (a)n as follows: N (cu) k (b) N, f(a)k = f(a)n, k = 1,,3.. (5) N! n= 1 where N is the number of longitudinal anti-resonance frequencies inclusive from N=1 used in the calculations. Using the single measurement technique to determine the resonance and antiresonance frequencies of the partially blocked duct it was found that taking N = 15 in equation (3) and (5) gave a sufficiently accurate approximation to make the reconstruction process valid. After successful blockage location and detection in ducts using the above technique, the author became interested in work by Sharp and Campbell 4, which showed that when the bore of a small pipe is reconstructed using acoustic pulse reflectrometry techniques a hole in the pipe wall gives rise to an apparent gradual expansion of the bore emanating from the position of that hole thus revealing its location. The current work describes application of the blockage location technique using resonance and anti-resonance frequency shifts as described above to the same problem. 3. SIZING OF A DUCT WALL HOLE If there is a hole of approximately circular cross-section in the wall of a duct with closed-open ends as illustrated in Figure 1 then the hole may be sized using an acoustic impedance model of the duct and the measured resonance and anti-resonance characteristics of the duct.
Driven closed end L 1 L Zclose Z 1 Z Z open d Hole in duct wall Z h Open end Figure 1: Impedance Model of Pipe with a Hole in the Wall The left hand section of the acoustic impedance model shown in Figure 1 is bounded by the plane of the driver and the plane of the hole while the right hand section is bounded by the plane of the hole and the plane of the open end. The complex impedance at the hole end of the left hand section is denoted Z 1 and the input impedance at the hole end of the right hand section by Z. Z h, the impedance of the wall orifice is given by: Z1 Z Zh = (6) Z Z1 The impedance Z 1 of the initial section of duct may be determined from the impedance Z closed of the fluid at the driver end using the impedance model for a section of duct with impedance terminations as follows: ρc Zclosed j tan( kl1 ) ρ c Sduct Z = (7) 1 S duct ρc j Zclosed tan( kl1 ) Sduct where L 1 is the longitudinal distance from the closed driver end of the duct to the orifice, S duct is the cross-sectional area of the duct, k is the acoustic wave number, ρ is the density of air and c is the speed of sound in air. If we wish to evaluate Z 1 then the fluid impedance Z closed at the driver is required. At resonance Z closed closely matches the very large impedance of the driver thus maximising power transmission into the duct. Thus Z closed becomes very large as the system becomes resonant and from equation (7): ρc j Z1 = (8) Sduct tan( k nl1 ) where k n is the wavenumber of the duct system at resonance. At anti-resonance Z closed is poorly matched to the large impedance of the closed end and therefore becomes very small as the system approaches anti-resonance. From equation (7): [ j k )] ρc Z1 = tan( ( a ) nl1 (9) Sduct where k (a)n is the anti-resonance wavenumber of the duct system.
Using a similar approach and using the standard equation for impedance of an unflanged pipe end, Z may be given by: 1 ρc Z = Sduct where r d is the cross-sectional radius of the duct. ( k rduct ) + j[ 0. 6( k rduct ) + tan( kl )] 4 1 [ 1 0. 6( k r ) tan( )] + ( ) tan( ) duct kl j k rduct kl 4 (10) The impedance of the orifice Z h may then be obtained by substituting (8) or (9) and (10) into (6). The impedance of an orifice of small dimensions relative to wavelength acting as a side branch in a duct wall is given by: ρ c k h ρ cl' k Z h h = + j (11) 4π π rh where r h is the orifice radius and k h is the complex wave number of propagation in the orifice, and α r h α k h = k - j (1) rh accounts for viscous losses in the hole and is given by: 1 µ κω α = + ( γ 1) (13) c ρc p Where µ is the gas viscosity (1.8*10-5 kg m -1 s -1 for air at 93 K), γ is the ratio of specific heats (1.4 for air), κ is the thermal conductivity (3*10-6 Wm -1 k -1 for air) and c p is the specific heat of air at constant pressure (1004.15 J Kg -1 K -1 for air). L' in equation (11) is equal to the thickness of the duct wall L h plus additional end corrections terms, and may be expressed as: r L ' = L + h h rh 1. 5 0. 58 (14) rd where r d is the radius of the main duct section. It is apparent that the second term inside the brackets in equation (14) will tend to zero as r h r becomes small in comparison to r d. Therefore, for values of h < 0. then we assume: rd L ' = Lh + 1.5r h (15) Now the imaginary part of equation (6) may be written as:
[ ] [( ) ( Im( Z )) ] ( Re( Z1 )) + ( Im( Z1 )) Im( Z1 ) Re( Z ) + [ Re( Z ) Re( Z )] + [ Im( Z ) Im( Z )] Im( Z ) Im( Z h ) = (16) 1 1 Substituting the imaginary parts of (8) or (9), (10) and (11) into equation (16) yields Im(Z h ). Now to obtain r h we substitute equations (1) and (15) into equation (11) and rearrange to give: π α Im( Zh ) r 1 5 rh Lh 0 ck h. = (17) ρ As α evaluated from (13) makes up less than 0.5% of the total term in rh even at near ultrasonic frequencies it may be deduced that viscous losses are insignificant and equation (17) simplifies to: π Im( Zh ) rh 1. 5rh Lh = 0 (18) ρck which may be solved in the standard way to yield r h. 4. EXPERIMENTAL ANALYSIS A m long plastic duct of 0.1 m diameter and wall thickness of 5.5 mm was used in the analysis. The duct resonance and anti-resonance characteristics were determined using Maximum Length Sequence (MLS) analysis. The substantial noise immunity and versatility of the MLS system has also been utilised to good effect in previous work by the author 1-3. The duct, which had a transverse mode cut-on frequency of approximately 1900 Hz, was excited by a piezo-electric driver attached to one end while the other end of the duct was left open. The excitation signal consisted of a 16384 point maximum length sequence of khz bandwidth and 8 khz sampling rate. The distorted MLS signal emanating from the driver was recorded by a microphone situated in the wall of the duct close to the driver enabling determination of the frequency transfer function of loudspeaker voltage to sound pressure in the duct. Four averages of the MLS signal were taken in each case giving a total measurement time of approximately 8 seconds. The MATLAB numerical processing package and a simple routine was utilised to select the resonance and anti-resonance frequencies of the duct from the measured data. Subsequently the resonance and anti-resonance frequency shifts of the duct with unperforated walls were computed from equations (3) and (5), and the resonance and anti-resonance value shifts were determined using equations () and (4). The apparent internal area function of the duct could then be generated using equation (1) thus revealing the location of the hole in the duct wall. The end corrected length of the duct L e in equation (1) was determined from the fundamental resonance frequency of the unperforated closed-open duct f 1 0 from equation (3) where: 1 c L e = () 4 0 f1 and the speed of sound c was determined as a function of temperature using a standard expression.
The initial fifteen resonance and anti-resonance values determined using equations () and (4) respectively were adequate for the internal area function reconstruction of the duct. All corresponding resonances and anti-resonances lay below the first duct cut-on frequency. 5. SIZING AND LOCATION OF HOLES 5.1. Location of Holes The internal bore sizing expression of equation (1) was used to locate the position of a hole in the duct wall from the measured transfer functions. The DC correction term a 0 in equation (1) was excluded as the notion of a blockage in the duct was now defunct. After completion of the reconstruction, the effect of the hole in the duct wall was immediately noticeable and its position could be ascertained. The reconstruction of the bore profile of the duct with a hole of 5 mm radius in the wall at of 1.4 m from the driver end of the duct is shown in Figure. A b (x)/a 0 (x) 1 0.5 0-0.5 a) -1 0 0. 0.4 0.6 0.8 1 1. 1.4 1.6 1.8 Axial Extent x (m) Figure : Reconstruction of Duct Area Function Showing Hole Location The substantial positive DC shift above the zero line is associated with an expansion in the duct (in earlier work negative DC shifts were found to occur in ducts with constrictions). The DC shift shown would be typical of a large expansion in the duct, which is effectively how the hole appears in the reconstruction. At the hole position a sharp inflection point occurs from which the curve drops steadily, crossing the zero line and continuing into what appears to be a flaring expansion towards the end of the duct. The point of inflection can be determined by a computational routine, which recognises the subsequent characteristic drop in the curve continuing below the zero line. 5.. Sizing of Circular Holes in the Duct Wall The hole sizing was undertaken using the technique set out above. Equations (8) and (9) were incorporated in separate analyses to see whether the measured wavenumber values at resonance or at anti-resonance k (a)n as determined from the duct transfer function gave the best sizing approximation. The values L 1 and L used describe the hole position in the longitudinal axis in Figure 1 and were determined computationally from the bore reconstruction generated using equation (1). The wavenumbers at first order resonance incorporated into equation (8) to determine Z 1 were generally found to be the most accurate for use the hole sizing process. Figure 3 shows the estimation of radius r h for holes at 1.4 m from the driver end using the above approach.
a) estimated radius of hole (m).. 0.005 0.004 0.003 0.00 0.001 0.000 0.001 0.00 0.003 0.004 0.005 radius of hole (m) Figure 3: Comparison of Actual and Predicted Hole Diameters Estimated Using Wavenumbers at First Order Resonance ( ) and Anti-resonance (ο). These estimated values were plotted against the actual values of the hole radius. Figure 3 shows estimations using equation (9). The estimated values shown in Figure 3 are highly accurate, all being within 10 % of the actual hole size. 5.3. Location and Sizing of Rectangular and Slit Shaped Holes in a Duct Wall The sizing technique was also applied to circumferential rectangular slits of varying aspect ratio (ratio of cross-sectional length to width). A rectangular slit was cut into the wall of the duct. A geometrical equivalent radius for the slit was computed from the slit cross-sectional area and compared to estimations of equivalent radius using the above technique. Figure 4 shows the equivalent radial estimations r he for circumferential slits of 1 mm breadth and 8mm and 8 mm length respectively determined using internal bore reconstruction and first resonance wavenumber k 1. These are plotted against the actual geometrical equivalent radii of the slits. The agreement between the actual and estimated values of r he were shown to be generally within 10% for the slits, with larger errors generally occurring when sizing slits of larger aspect ratio. This error occurs because the end correction of the lumped element of air bounded by the slit is dominated by the shorter slit dimension. Thus the end correction is shorter than for a circular hole of identical area, which causes the technique to oversize the areas of slit. This error is increases with increased slit aspect ratio.
aspect ratio (width of crack: length of crack) estimated equivalent radius of crack (m).. 1:8 1:8 0.004 0.003 0.00 0.001 0.000 0.000 0.001 0.00 0.003 0.004 equivalent radius of crack (m) Figure 4. Acoustically determined values of geometric equivalent radius r he for slit shaped holes of 1 mm breadth and varying aspect ratio as a function of actual r he value (with 10% vertical error bars). Slits positioned at 0.454 m from driver. 6. CONCLUSIONS A technique has been developed for the detection, location and sizing of small apertures in a duct wall using the resonance and anti-resonance frequencies of the duct determined from a single transfer function measurement. The technique reconstructs the apparent internal area function of the duct from which the position of the hole may be ascertained by eye or detected automatically using a simple computational routine. Assuming the hole is of circular crosssection then, following its location, it may be sized using an impedance model of the duct. The location of the hole using this method is shown to be highly accurate and subsequent sizing of the hole is found to be most accurate using the measured first order resonance wave number of the duct. The detection, location and sizing technique has also been successfully applied to slit shaped holes where the sizing process generates a geometrical equivalent radius for the slit. 7. REFERENCES 1. M.H.F. de Salis and D.J. Oldham 001, The development of a rapid single spectrum method for determining the blockage characteristics of a Finite length duct, J. Sound Vib. 43(4), 65-640.. M. H. F. de Salis and D. J. Oldham, 1999, Determination of the blockage area function of a finite duct from a single pressure response measurement, J. Sound Vib. 1(1), 180-186. 3. M. H. F. de Salis and D. J. Oldham, 000, A rapid technique to determine the internal area function of finite-length ducts using maximum length sequence analysis, J. Acoust. Soc. Am. 108 (1), 44-5. 4. D.B. Sharp and D.M. Campbell, 1997, Leak detection in pipes using acoustic pulse reflectometry, Acustica 83 (3) 560-566.