D. Elad M. Sahar J. M. Avidor S. Einav Biomedical Engineering Program, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978, Israel
Steady Flow Through Collapsible Tubes: Measurements of Flow and Geometry Compliant tubes attain a complex three-dimensional geometry when the external pressure exceeds the internal pressure and the tube is partially collapsed. A new technique for remote measurement of dynamic surfaces was applied to classical experiments with collapsible tubes. This work presents measurements of the threedimensional structure of the tube as well as pressure and flow measurements during static loading and during steady-state fluid flow. Results are shown for two tubes of the same material and internal diameter but with different wall thicknesses. The measured tube laws compare well with previously published data and suggest the possible existence of a similarity tube law. The steady flow measurements did not compare well with the one-dimensional theoretical predictions.
Introduction Fluid flow through collapsible tubes is a complex threedimensional nature due to the coupled interaction between the tube-wall mechanics and the pressure distribution of the flowing fluid. It is usually used to simulate biological flows such as blood flow in arteries or veins and air flow in the bronchial airways. So far, only one-dimensional models have been developed to investigate the essentials of fluid flow through collapsible tubes and the effect of various parameters [13, 17, 19, 23]. In these models, the equations that describe the displacement of the tube wall (equilibrium + constitutive) are replaced by a "tube law" that relates the tube cross-sectional area to the pressure difference across the tube wall. These works draw upon the analogy between incompressible fluid flow through a collapsible tube and compressible fluid flow in a rigid tube (gas dynamics). Experimental investigations of flow through collapsible tubes have been limited, because invasive measurements alter the flow condition. In particular, it is impossible to measure the tube geometry invasively since any contact with a measuring device alters the dynamic equilibrium of the tube wall. Thus, there is no experimental evidence on the exact three-dimensional geometry of the tube under steady or unsteady flow conditions. Kececioglu et al. [15] and Bertram [2] used the electrical impedance method to measure cross-sectional areas of a collapsed tube during static loading or various flow conditions, providing data suitable only for one-dimensional analysis. Bertram and Ribreau [3] used the transformer principle to measure cross-sectional areas with a metallic wire embedded in its wall. However, a deeper insight into local phenomena, especially those that occur during growing disturbances such as wave steepening, elastic jumps or self-excited oscillations, requires experimental information on the three-dimensional
geometry of the tube along with data on the fluid flow characteristics. Some investigators have succeeded in measuring the shape of a single cross-section of the tube, subjected to static loading, by methods of X-ray cinefluorography [18] or ultrasound imaging [3, 21], However, the resolution of these methods is limited and only single cross-sectional contours were generated. The desire to measure the geometry of the trachea during forced expiration stimulated a few investigations. A setup composed of two camera-projector systems and a reference frame with twelve control points was developed on the basis of photogrammetric principles to measure an artificial trachea [20, 24]. A coded rectangular grid was projected on the tube and photographed with a camera positioned perpendicular to the projector. However, measurements of collapsible tubes during static or dynamic loading have not yet been reported. In other works, existing medical imaging systems such as cine-computer tomography [25] or magnetic resonance imaging [16] have been used to measure the geometry of a trachea during forced expiration maneuvers. Recently, we have independently developed a remote sensing technique for measurements of biomedical surfaces subjected to static or dynamic loading [8, 22]. This new technique avoids the complex mathematical computation of existing stereophotogrammetric methods [1] and yet yields accurate measurements of three-dimensional surfaces even when the optical path traverses different media. It can be easily applied to measure the shape of a collapsible tube installed in a liquid-filled chamber for in vitro experiments. Here, we present three-dimensional measurements of a collapsible tube which is subjected either to static loading or to steady fluid flow through it. 2
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division August 6, 1991; revised manuscript received September 30, 1991.
Experimental Apparatus
2.1 General Description. The goal of the present investigation was to measure the geometry of a collapsible latex
84 / Vol. 114, FEBRUARY 1992
Transactions of the ASME
Copyright © 1992 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
/;}////// Section A-A
*^~-
////////s///
CONTROL
COLLAPSIBLE' TUBE
mmmm
^Jssssmmmmmsm View C-C
View B-B
Fig. 1 Schematic description of the experimental setup. Section A-A describes the relative position of the side camera and the four control planes. Section B-B shows the coded raster as seen from above. Section C-C shows the image as depicted by the CCD-TV camera.
tube while fluid flows through it. The geometry measurements were accompanied with traditional measurements of flow rate and transmural pressures. The experiments were conducted in a flow circuit which was controlled by a computer and equipped with optical components for remote sensing of the tube geometry. Details of the technique of surface measurement appear elsewhere [8, 22]. For the convenience of the reader we describe it briefly. 2.2 Flow Circuit. The flow circuit was constructed with the tube mounted in a water-filled rectangular transparent test chamber to eliminate gravity effects (Fig. 1). Water flow from a constant-head tank enters the flexible tube through a computer-operated flow-adjusting valve, and exits through a second valve into a downstream reservoir. The adjusting valves were constructed from commercial butterfly valves, operated by step motors and driven by a computer. Various flow rates were obtained by varying the pressure in the chamber and/or the resistance to flow at the valves. The flow rate was measured by an electromagnetic flowmeter (Carolina FM501) with a 26 mm probe (EP680). The pressure in the water-filled test chamber, externally to the latex tube (Pe), was controlled by a vertically moving tank. The internal pressure (P) was assumed uniform over the crosssection and was measured at the tube axis via a movable catheter. The transmural pressure across the tube wall (internal minus external pressure) was determined from P,m = P-Pe. The exact location of the catheter was determined by a linear variable differential transformer (LVDT; Bakara ST50). The catheter was made of a stainless steel tube: 1.2 and 1.75 mm inside and outside diameters, respectively. It was blocked at the middle and two measuring holes of 0.5 mm diameter were drilled, one on each side of the blockage, 50 mm apart. Each end of the catheter was connected to a pressure transducer Journal of Biomechanical Engineering
(Validyne DP15-36; ±35 KPa), thus permitting simultaneous measurements at two locations. Experiments were conducted with two latex tubes (manufactured by dipping) each of 27 mm internal (unstressed) diameter with different wall thicknesses, 0.8 and 1.6 mm. The final length of the tube in the test chamber was 490 mm. At the downstream end of the tube, 70 mm before the fixation, the tube was manufactured with a gradual wall thickening (up to 5 mm) to avoid formation of self-excited oscillations under most conditions of steady flows. 2.3 Measurement of the Tube Geometry. The tube is assumed symmetric about the horizontal plane, thus, it is sufficient to measure its upper surface by using the noncontact technique we have developed [8]. The upper surface of the tube was vertically screened by a coded raster (grid of parallel lines) so that each raster line coincides with the tube crosssection taken perpendicular to the tube axis (Fig. 1; view BB). The resulting image was photographed with a CCD-TV camera mounted 1.10 m above the tube and at an inclination of about 45 deg (Fig. 1). The camera image, taken at an angle to the tube axis, consists of the same raster lines but distorted due to their different heights on the object surface and control surfaces (Fig. 1; view C-C). ' Three control surfaces, made of aluminum and 10 mm wide, were installed in the test chamber on both sides of the tube for reconstruction purposes (Fig. 1; section A-A). Two surfaces were machined to form two parallel stairs with a vertical separation (height) of 20 mm and installed to the left of the tube. Another surface was mounted to the right of the tube so that it was coplanar with the upper surface on the left side. The two upper coplanar surfaces were marked with six control points for calibration purposes (Fig. 1; views B-B and C-C). The control surfaces were mounted along the tube so that the FEBRUARY 1992, Vol. 114/85
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lower left surface coincided with the horizontal plane that contained the nominal center-line of the tube. Below the top right surface, a fourth surface was constructed for calibration of measurements taken from a side, as will be discussed later. In the present study, 100 mm of the tube length at midsection was measured. It was screened by 30 lines, thus, producing data each 3 mm. The photographed images were either recorded on a professional video tape or digitized directly to a 512x512 pixel image of 256 grey levels and transferred to a Microvax for further processing. The basic assumption of the measuring technique is that the camera-tube distance is much larger than the tube radius (about 75:1), and as a result, linear relationships are assumed between the heights of the tube surface and their projections on the image plane. Accordingly, the digitized image was straightened to correct only for distortions caused by optical parallax and the relative angle between the camera and the horizontal plane [22]. In the final processing stages, the control points were used to calibrate the data within the image plane, and the vertical separation between the control planes, to calibrate the heights of the upper surface of the tube. In experiments with collapsible tubes, the tube center-line is not maintained fixed in the horizontal plane of symmetry during all the possible equilibrium conditions. Since we measured only the upper half of the tube, it was necessary to evaluate the horizontal axis of symmetry at each longitudinal location. For this purpose, another CCD-TV camera was positioned on the right side of the tube in the horizontal plane of symmetry at a distance of 1.50 m from the tube. A fourth control plane was installed below the top right plane to provide a known side reference (Fig. 1; section A-A). The tube was simultaneously photographed from both the top and the side. The lowest point of the tube, at each longitudinal location, was measured by the side camera and was compared with the highest point from the top measurements to yield the exact location of the tube axis. The performance and accuracy of the geometry measuring system were checked by measuring the surface of a known cylindrical phantom with a diameter of 30 mm. The measured data at five sections along the measured zone (approximately equally spaced) are shown in Fig. 2(a) along with the exact contour of the phantom. The data was obtained from a section of about 112 deg rather than the desired 180 deg. The limited angle of view may be attributed to two reasons: 1) the cameraobject distance is not infinity; and 2) the raster lines become very close to each other (actually in contact) towards the tube edges, which prohibits differentiation between the lines. When the tube is collapsed, the angle of view is largely increased (Fig. 3). The differences between the measured data and the exact values are in the range of - 0.72 to 0.041 mm as depicted in Fig. 2(b). The averaged deviation for all the 5 section was -0.258 mm (1.06 pixel in the digital data) with a standard deviation of ±0.109 mm (±0.44 pixel). The results of this evaluation will be discussed later concerning the correction of the measured data. 2.4 Data Acquisition. The pressures, flow rate and catheter position were measured with a personal computer equipped with an analog-to-digital converter (Tecmar TM-40). In the present work we measured cases of static loading and steady state flow, hence we sampled the data at a frequency of 10 Hz during 3 seconds and later averaged. The accuracy of the pressure and flow measurements was ± 1 percent and that of the LVDT ±0.4 percent. The geometry measurements provided data, X, Y, Z in mm, of the outer cross-sectional contours of the tube at each raster line along the measuring zone. However, in one-dimensional analysis of flow through collapsible tubes, the internal crosssectional area is of interest. Thus, the measured geometry of the tube needs further manipulation. First, we corrected the 86 / Vol. 114, FEBRUARY 1992
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X[mm] Fig. 2 Comparison between measured contours and exact contours of the cylindrical phantom: (a) real measurements; (b) relative error
Fig. 3 Schematic description of corrections for wall thickness and missing data
measured data for the accuracy of the measuring system as has been discussed earlier. The difference between the phantom cross-sectional area and that of each of the measured contours was within 1 percent of the averaged error (Fig. 2(b)) multiplied by the tube diameter. Accordingly, we added to the area below each of the measured contours the area of a rectangle obtained by multiplying the number of pixels along the width of the cross-section by 1.06 and by the scaling factor for the particular raster line. Then, we subtracted the wall thickness perpendicular to the measured surface to yield the internal cross-section of the tube. The wall thickness is assumed constant both at collapsed and inflated states. This is justified in inflated tubes by the fact that circumferential as well as longitudinal extensions are very small (less than 2-3 percent), and thus, changes in wall thickness are negligible. Finally, we corrected the measurements for the data missing due to the limited angle of view. The missing areas at the outer bound of the tube cross-section were considered as triangles whose bases were evaluated by manual measurements (with a mouse) of the external width of the projected tube (e.g., outside diameter of a circular tube, see Fig. 3) from the straightened image. From this measure we subtracted the wall thickness and the total width (in Xdirection) which was measured with the remote technique (Section 2.3) in order to obtain the bases of the triangles on each side (Fig. 3). The corrections for wall thickness and missing "triangles" were evaluated on a theoretical tube; 30 mm outside diameter and 1.6 mm wall thickness. The computed internal cross-sectional area showed a relative deviation of - 0.86 percent from the exact value, which is small compared with the measuring error of the system. The accuracy of the complete procedure that evaluates the internal cross sectional area from the threeTransactions of the AS ME
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dimensional measurements of the outer surface was analyzed by assuming that the cylindrical phantom is a tube with wall thickness of 1.6 mm and using the measured contours of Fig. 2(a). The resulting internal cross-sectional area of this tube was 3 percent smaller than the exact area. As the tube is in a further collapsed stage, the absolute error decreases (Fig. 3), however, the cross-sectional area decreases as well. 3
Results and Discussion Experiments were performed with two latex tubes of different wall thicknesses (Table 1). The tubes were installed in the test chamber with a slight axial extension (1-3 percent) to avoid longitudinal bending mainly during inflation states. The measuring zone, chosen to be 100 mm long at the middle of the tube, was far enough (170 mm) from each end that longitudinal bending could be ignored. The tube geometry was measured at different transmural pressures and then accompanied by experiments of steady-state flow in various collapsed conditions. 3.1
Tube Law.
The tube law was obtained by evaluating
Table 1 Geometric and material parameters as well as the coefficients of the fitted curves of tube laws for the latex tubes Thin tube Thick tube Geometric and Material Parameters: 1.6 h0 [mm] 0.8 L [cm] 2 10 10 A0 [cm ] 5.72 6.64 /•„ [cm] 1.35 1.45 e= ^ [ % ]
3.7
1.1
E [cm H 2 0] Kbp [cm H 2 0]
18,284 0.423
20,287 3.028
Coefficients for Plm = n, n2 Kp[cmH20]
Kp(^-a~"^): 44 1.3 1.387
Coefficients for P„„ = Kp{e"'(a~c'o) - e-"^a~- 0 . The coefficient terial they are made of, as long as their wall-thickness to radius C is used as a constraint that ensures Ptm = 0 for a = 1. The fitted curves were evaluated as above and are depicted as dashed ratio is within a defined range. The theoretical model that leads to the similarity tube law lines in Fig. 6 while the best fitted coefficients for both tubes for elastic tubes assumes a constitutive relation in which the are summarized in Table 1. Unfortunately, the set of coeffibending moment is proportional to the deviation of the cur- cients for the thin-walled tube (Fig. 6(b)) which reproduces the vature of the cross-section from its value in the circular state experimental data during inflation did not reproduce the data 30
88 / Vol. 114, FEBRUARY 1992
Transactions of the ASME
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-300
|
Fig. 6 Theoretical curves in comparison with experimental data of the averaged tube law (' +'). (a) thick-wall tube, (b) thin-wall tube. Continuous lines: Elad's et al. [6] model Plm= K^oTi -a~"2). Dashed lines: hyperbolic sine model /ye"i < ""°o ) - e",2{"~"")- C]. The upper panels show the wave speed, c, as derived from each theoretical curve, />,„,.
during collapse. As this work is more concerned with the collapsed region, the set of coefficients which produced the best fit to this region was selected. The theoretical tube laws were also used to evaluate the speed of propagation of small area perturbations. The wave speed was computed from (?= (a/p) (dP,m/da) for each of the fitted tube laws and the results are depicted in the upper panels of Fig. 6. The hyperbolic functions, that produce a very good fit to the measured data at the rapid collapse region, yield very low values at this region; 0.65 and 0.17 cm/s for the thick and thin wall tube, respectively. 3.2 Steady State Flow. The one-dimensional field of steady fluid flow through collapsible tubes has been predicted by several investigators (e.g., Shapiro [23]). This, however, was based on the assumption that a tube law model replaces the actual three-dimensional mechanical characteristics of the tube. We therefore selected to apply the three-dimensional measuring technique to experiments on steady flow through the latex tubes describe above. In each experiment we measured the flow rate, the tube geometry and the transmural pressure distribution along the central 100 mm section of the tube. The variation of the internal static pressure along the tube was very small and as a result the variations in the transmural pressure are smaller than the accuracy of the large range pressure transducer (Validyne DP15-36; ±35 KPa). To overcome this difficulty we determined the internal static pressure (P) at ten equally spaced locations with respect to that at a reference point at the entrance to the test chamber using a very sensitive and accurate differential pressure transducer (Validyne DP10310; ±86 Pa). The transmural pressure (P-Pe) distribution was evaluated from these measurements and from the transmural pressure at the reference point which was measured with the large range and less sensitive differential pressure transducer. As a result, the error of pressure measurements (5P) was determined by the accuracy of the large range transducer and was relatively large. The axial distributions of the measured data of transmural pressure, Plm, cross-sectional area, A, and fluid velocity, U=Q/ A, are shown in Figs. 7 and 8. The test-chamber pressures (Pe), the flow rates (Q, the estimated error of pressure measurements (8P) and a typical Reynolds number (Re) for each of the experiments are summarized in Table 2. The surfaces of the tubes in each of the steady flow experiments are also shown in Figs. 7 and 8. They were plotted by a commercial Journal of Biomechanical Engineering
30
:*
* t
Thick-Wall Tube H
i M t t t *
15
, 1
1
I < > I
x + + + +
Q = 122.9 ml/s (+) -4 800 * * * *
+ + + +
-
§ 400
Q = II9.2 mJ/s(*) "0.0
0.5
1.0
Fig. 7 Measurements of steady fluid flows through the thick-wall tube. (+) represent values for Q = 122.9 ml/s, (*) represent values for 0 = 119.2 ml/s. The dimensionless length is J = x/L, where L is the length of the measured region.
package that computes the surface from discrete data (e.g., contours of given cross-sections) and draws the surface with a predefined number of longitudinal and circumferential grid lines. The surfaces in Figs. 7 and 8 were generated from measured contours of eleven cross-sections along the measuring zone (L = 100 mm). In the present work, exact contours (about 35) were obtained at an axial resolution of 3 mm. This can be improved, if necessary, by increasing the number of raster lines projected on the measuring region. In some cases the tube was not symmetric about the horizontal plane and rotation of the cross-section can be observed (e.g., Fig. 8; Q = 38.8 ml/ FEBRUARY 1992, Vol. 114 / 89
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Thin-Wall Tube
x E iH.
Table 2 Reference parameters for the steady flows. Re = UDe/ v where De is the hydraulic diameter (4A/s), v is the kinematic viscosity, and s is the cross-sectional perimeter (assumed constant during collapse states)
Pe [cm H2OJ Q [ml/s] bP [cm H 2 0] Re(£ = 0) fT (Blasius)
'
Thin tube Case II Case I 59.03 57.04 43.2 38.8 0.285 0.295 2018 1790 0.012 0.012
Thick tube Case III Case IV 11.48 11.88 122.9 119.2 0.0594 0.0574 5,325 5165 0.009 0.009
-2-
E QT
1
Steady Flow Through Collapsible Tubes: Measurements of Flow and Geometry Compliant tubes attain a complex three-dimensional geometry when the external pressure exceeds the internal pressure and the tube is partially collapsed. A new technique for remote measurement of dynamic surfaces was applied to classical experiments with collapsible tubes. This work presents measurements of the threedimensional structure of the tube as well as pressure and flow measurements during static loading and during steady-state fluid flow. Results are shown for two tubes of the same material and internal diameter but with different wall thicknesses. The measured tube laws compare well with previously published data and suggest the possible existence of a similarity tube law. The steady flow measurements did not compare well with the one-dimensional theoretical predictions.
Introduction Fluid flow through collapsible tubes is a complex threedimensional nature due to the coupled interaction between the tube-wall mechanics and the pressure distribution of the flowing fluid. It is usually used to simulate biological flows such as blood flow in arteries or veins and air flow in the bronchial airways. So far, only one-dimensional models have been developed to investigate the essentials of fluid flow through collapsible tubes and the effect of various parameters [13, 17, 19, 23]. In these models, the equations that describe the displacement of the tube wall (equilibrium + constitutive) are replaced by a "tube law" that relates the tube cross-sectional area to the pressure difference across the tube wall. These works draw upon the analogy between incompressible fluid flow through a collapsible tube and compressible fluid flow in a rigid tube (gas dynamics). Experimental investigations of flow through collapsible tubes have been limited, because invasive measurements alter the flow condition. In particular, it is impossible to measure the tube geometry invasively since any contact with a measuring device alters the dynamic equilibrium of the tube wall. Thus, there is no experimental evidence on the exact three-dimensional geometry of the tube under steady or unsteady flow conditions. Kececioglu et al. [15] and Bertram [2] used the electrical impedance method to measure cross-sectional areas of a collapsed tube during static loading or various flow conditions, providing data suitable only for one-dimensional analysis. Bertram and Ribreau [3] used the transformer principle to measure cross-sectional areas with a metallic wire embedded in its wall. However, a deeper insight into local phenomena, especially those that occur during growing disturbances such as wave steepening, elastic jumps or self-excited oscillations, requires experimental information on the three-dimensional
geometry of the tube along with data on the fluid flow characteristics. Some investigators have succeeded in measuring the shape of a single cross-section of the tube, subjected to static loading, by methods of X-ray cinefluorography [18] or ultrasound imaging [3, 21], However, the resolution of these methods is limited and only single cross-sectional contours were generated. The desire to measure the geometry of the trachea during forced expiration stimulated a few investigations. A setup composed of two camera-projector systems and a reference frame with twelve control points was developed on the basis of photogrammetric principles to measure an artificial trachea [20, 24]. A coded rectangular grid was projected on the tube and photographed with a camera positioned perpendicular to the projector. However, measurements of collapsible tubes during static or dynamic loading have not yet been reported. In other works, existing medical imaging systems such as cine-computer tomography [25] or magnetic resonance imaging [16] have been used to measure the geometry of a trachea during forced expiration maneuvers. Recently, we have independently developed a remote sensing technique for measurements of biomedical surfaces subjected to static or dynamic loading [8, 22]. This new technique avoids the complex mathematical computation of existing stereophotogrammetric methods [1] and yet yields accurate measurements of three-dimensional surfaces even when the optical path traverses different media. It can be easily applied to measure the shape of a collapsible tube installed in a liquid-filled chamber for in vitro experiments. Here, we present three-dimensional measurements of a collapsible tube which is subjected either to static loading or to steady fluid flow through it. 2
Contributed by the Bioengineering Division for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING. Manuscript received by the Bioengineering Division August 6, 1991; revised manuscript received September 30, 1991.
Experimental Apparatus
2.1 General Description. The goal of the present investigation was to measure the geometry of a collapsible latex
84 / Vol. 114, FEBRUARY 1992
Transactions of the ASME
Copyright © 1992 by ASME Downloaded From: http://biomechanical.asmedigitalcollection.asme.org/ on 01/28/2016 Terms of Use: http://www.asme.org/about-asme/terms-of-use
/;}////// Section A-A
*^~-
////////s///
CONTROL
COLLAPSIBLE' TUBE
mmmm
^Jssssmmmmmsm View C-C
View B-B
Fig. 1 Schematic description of the experimental setup. Section A-A describes the relative position of the side camera and the four control planes. Section B-B shows the coded raster as seen from above. Section C-C shows the image as depicted by the CCD-TV camera.
tube while fluid flows through it. The geometry measurements were accompanied with traditional measurements of flow rate and transmural pressures. The experiments were conducted in a flow circuit which was controlled by a computer and equipped with optical components for remote sensing of the tube geometry. Details of the technique of surface measurement appear elsewhere [8, 22]. For the convenience of the reader we describe it briefly. 2.2 Flow Circuit. The flow circuit was constructed with the tube mounted in a water-filled rectangular transparent test chamber to eliminate gravity effects (Fig. 1). Water flow from a constant-head tank enters the flexible tube through a computer-operated flow-adjusting valve, and exits through a second valve into a downstream reservoir. The adjusting valves were constructed from commercial butterfly valves, operated by step motors and driven by a computer. Various flow rates were obtained by varying the pressure in the chamber and/or the resistance to flow at the valves. The flow rate was measured by an electromagnetic flowmeter (Carolina FM501) with a 26 mm probe (EP680). The pressure in the water-filled test chamber, externally to the latex tube (Pe), was controlled by a vertically moving tank. The internal pressure (P) was assumed uniform over the crosssection and was measured at the tube axis via a movable catheter. The transmural pressure across the tube wall (internal minus external pressure) was determined from P,m = P-Pe. The exact location of the catheter was determined by a linear variable differential transformer (LVDT; Bakara ST50). The catheter was made of a stainless steel tube: 1.2 and 1.75 mm inside and outside diameters, respectively. It was blocked at the middle and two measuring holes of 0.5 mm diameter were drilled, one on each side of the blockage, 50 mm apart. Each end of the catheter was connected to a pressure transducer Journal of Biomechanical Engineering
(Validyne DP15-36; ±35 KPa), thus permitting simultaneous measurements at two locations. Experiments were conducted with two latex tubes (manufactured by dipping) each of 27 mm internal (unstressed) diameter with different wall thicknesses, 0.8 and 1.6 mm. The final length of the tube in the test chamber was 490 mm. At the downstream end of the tube, 70 mm before the fixation, the tube was manufactured with a gradual wall thickening (up to 5 mm) to avoid formation of self-excited oscillations under most conditions of steady flows. 2.3 Measurement of the Tube Geometry. The tube is assumed symmetric about the horizontal plane, thus, it is sufficient to measure its upper surface by using the noncontact technique we have developed [8]. The upper surface of the tube was vertically screened by a coded raster (grid of parallel lines) so that each raster line coincides with the tube crosssection taken perpendicular to the tube axis (Fig. 1; view BB). The resulting image was photographed with a CCD-TV camera mounted 1.10 m above the tube and at an inclination of about 45 deg (Fig. 1). The camera image, taken at an angle to the tube axis, consists of the same raster lines but distorted due to their different heights on the object surface and control surfaces (Fig. 1; view C-C). ' Three control surfaces, made of aluminum and 10 mm wide, were installed in the test chamber on both sides of the tube for reconstruction purposes (Fig. 1; section A-A). Two surfaces were machined to form two parallel stairs with a vertical separation (height) of 20 mm and installed to the left of the tube. Another surface was mounted to the right of the tube so that it was coplanar with the upper surface on the left side. The two upper coplanar surfaces were marked with six control points for calibration purposes (Fig. 1; views B-B and C-C). The control surfaces were mounted along the tube so that the FEBRUARY 1992, Vol. 114/85
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lower left surface coincided with the horizontal plane that contained the nominal center-line of the tube. Below the top right surface, a fourth surface was constructed for calibration of measurements taken from a side, as will be discussed later. In the present study, 100 mm of the tube length at midsection was measured. It was screened by 30 lines, thus, producing data each 3 mm. The photographed images were either recorded on a professional video tape or digitized directly to a 512x512 pixel image of 256 grey levels and transferred to a Microvax for further processing. The basic assumption of the measuring technique is that the camera-tube distance is much larger than the tube radius (about 75:1), and as a result, linear relationships are assumed between the heights of the tube surface and their projections on the image plane. Accordingly, the digitized image was straightened to correct only for distortions caused by optical parallax and the relative angle between the camera and the horizontal plane [22]. In the final processing stages, the control points were used to calibrate the data within the image plane, and the vertical separation between the control planes, to calibrate the heights of the upper surface of the tube. In experiments with collapsible tubes, the tube center-line is not maintained fixed in the horizontal plane of symmetry during all the possible equilibrium conditions. Since we measured only the upper half of the tube, it was necessary to evaluate the horizontal axis of symmetry at each longitudinal location. For this purpose, another CCD-TV camera was positioned on the right side of the tube in the horizontal plane of symmetry at a distance of 1.50 m from the tube. A fourth control plane was installed below the top right plane to provide a known side reference (Fig. 1; section A-A). The tube was simultaneously photographed from both the top and the side. The lowest point of the tube, at each longitudinal location, was measured by the side camera and was compared with the highest point from the top measurements to yield the exact location of the tube axis. The performance and accuracy of the geometry measuring system were checked by measuring the surface of a known cylindrical phantom with a diameter of 30 mm. The measured data at five sections along the measured zone (approximately equally spaced) are shown in Fig. 2(a) along with the exact contour of the phantom. The data was obtained from a section of about 112 deg rather than the desired 180 deg. The limited angle of view may be attributed to two reasons: 1) the cameraobject distance is not infinity; and 2) the raster lines become very close to each other (actually in contact) towards the tube edges, which prohibits differentiation between the lines. When the tube is collapsed, the angle of view is largely increased (Fig. 3). The differences between the measured data and the exact values are in the range of - 0.72 to 0.041 mm as depicted in Fig. 2(b). The averaged deviation for all the 5 section was -0.258 mm (1.06 pixel in the digital data) with a standard deviation of ±0.109 mm (±0.44 pixel). The results of this evaluation will be discussed later concerning the correction of the measured data. 2.4 Data Acquisition. The pressures, flow rate and catheter position were measured with a personal computer equipped with an analog-to-digital converter (Tecmar TM-40). In the present work we measured cases of static loading and steady state flow, hence we sampled the data at a frequency of 10 Hz during 3 seconds and later averaged. The accuracy of the pressure and flow measurements was ± 1 percent and that of the LVDT ±0.4 percent. The geometry measurements provided data, X, Y, Z in mm, of the outer cross-sectional contours of the tube at each raster line along the measuring zone. However, in one-dimensional analysis of flow through collapsible tubes, the internal crosssectional area is of interest. Thus, the measured geometry of the tube needs further manipulation. First, we corrected the 86 / Vol. 114, FEBRUARY 1992
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X[mm] Fig. 2 Comparison between measured contours and exact contours of the cylindrical phantom: (a) real measurements; (b) relative error
Fig. 3 Schematic description of corrections for wall thickness and missing data
measured data for the accuracy of the measuring system as has been discussed earlier. The difference between the phantom cross-sectional area and that of each of the measured contours was within 1 percent of the averaged error (Fig. 2(b)) multiplied by the tube diameter. Accordingly, we added to the area below each of the measured contours the area of a rectangle obtained by multiplying the number of pixels along the width of the cross-section by 1.06 and by the scaling factor for the particular raster line. Then, we subtracted the wall thickness perpendicular to the measured surface to yield the internal cross-section of the tube. The wall thickness is assumed constant both at collapsed and inflated states. This is justified in inflated tubes by the fact that circumferential as well as longitudinal extensions are very small (less than 2-3 percent), and thus, changes in wall thickness are negligible. Finally, we corrected the measurements for the data missing due to the limited angle of view. The missing areas at the outer bound of the tube cross-section were considered as triangles whose bases were evaluated by manual measurements (with a mouse) of the external width of the projected tube (e.g., outside diameter of a circular tube, see Fig. 3) from the straightened image. From this measure we subtracted the wall thickness and the total width (in Xdirection) which was measured with the remote technique (Section 2.3) in order to obtain the bases of the triangles on each side (Fig. 3). The corrections for wall thickness and missing "triangles" were evaluated on a theoretical tube; 30 mm outside diameter and 1.6 mm wall thickness. The computed internal cross-sectional area showed a relative deviation of - 0.86 percent from the exact value, which is small compared with the measuring error of the system. The accuracy of the complete procedure that evaluates the internal cross sectional area from the threeTransactions of the AS ME
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dimensional measurements of the outer surface was analyzed by assuming that the cylindrical phantom is a tube with wall thickness of 1.6 mm and using the measured contours of Fig. 2(a). The resulting internal cross-sectional area of this tube was 3 percent smaller than the exact area. As the tube is in a further collapsed stage, the absolute error decreases (Fig. 3), however, the cross-sectional area decreases as well. 3
Results and Discussion Experiments were performed with two latex tubes of different wall thicknesses (Table 1). The tubes were installed in the test chamber with a slight axial extension (1-3 percent) to avoid longitudinal bending mainly during inflation states. The measuring zone, chosen to be 100 mm long at the middle of the tube, was far enough (170 mm) from each end that longitudinal bending could be ignored. The tube geometry was measured at different transmural pressures and then accompanied by experiments of steady-state flow in various collapsed conditions. 3.1
Tube Law.
The tube law was obtained by evaluating
Table 1 Geometric and material parameters as well as the coefficients of the fitted curves of tube laws for the latex tubes Thin tube Thick tube Geometric and Material Parameters: 1.6 h0 [mm] 0.8 L [cm] 2 10 10 A0 [cm ] 5.72 6.64 /•„ [cm] 1.35 1.45 e= ^ [ % ]
3.7
1.1
E [cm H 2 0] Kbp [cm H 2 0]
18,284 0.423
20,287 3.028
Coefficients for Plm = n, n2 Kp[cmH20]
Kp(^-a~"^): 44 1.3 1.387
Coefficients for P„„ = Kp{e"'(a~c'o) - e-"^a~- 0 . The coefficient terial they are made of, as long as their wall-thickness to radius C is used as a constraint that ensures Ptm = 0 for a = 1. The fitted curves were evaluated as above and are depicted as dashed ratio is within a defined range. The theoretical model that leads to the similarity tube law lines in Fig. 6 while the best fitted coefficients for both tubes for elastic tubes assumes a constitutive relation in which the are summarized in Table 1. Unfortunately, the set of coeffibending moment is proportional to the deviation of the cur- cients for the thin-walled tube (Fig. 6(b)) which reproduces the vature of the cross-section from its value in the circular state experimental data during inflation did not reproduce the data 30
88 / Vol. 114, FEBRUARY 1992
Transactions of the ASME
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-300
|
Fig. 6 Theoretical curves in comparison with experimental data of the averaged tube law (' +'). (a) thick-wall tube, (b) thin-wall tube. Continuous lines: Elad's et al. [6] model Plm= K^oTi -a~"2). Dashed lines: hyperbolic sine model /ye"i < ""°o ) - e",2{"~"")- C]. The upper panels show the wave speed, c, as derived from each theoretical curve, />,„,.
during collapse. As this work is more concerned with the collapsed region, the set of coefficients which produced the best fit to this region was selected. The theoretical tube laws were also used to evaluate the speed of propagation of small area perturbations. The wave speed was computed from (?= (a/p) (dP,m/da) for each of the fitted tube laws and the results are depicted in the upper panels of Fig. 6. The hyperbolic functions, that produce a very good fit to the measured data at the rapid collapse region, yield very low values at this region; 0.65 and 0.17 cm/s for the thick and thin wall tube, respectively. 3.2 Steady State Flow. The one-dimensional field of steady fluid flow through collapsible tubes has been predicted by several investigators (e.g., Shapiro [23]). This, however, was based on the assumption that a tube law model replaces the actual three-dimensional mechanical characteristics of the tube. We therefore selected to apply the three-dimensional measuring technique to experiments on steady flow through the latex tubes describe above. In each experiment we measured the flow rate, the tube geometry and the transmural pressure distribution along the central 100 mm section of the tube. The variation of the internal static pressure along the tube was very small and as a result the variations in the transmural pressure are smaller than the accuracy of the large range pressure transducer (Validyne DP15-36; ±35 KPa). To overcome this difficulty we determined the internal static pressure (P) at ten equally spaced locations with respect to that at a reference point at the entrance to the test chamber using a very sensitive and accurate differential pressure transducer (Validyne DP10310; ±86 Pa). The transmural pressure (P-Pe) distribution was evaluated from these measurements and from the transmural pressure at the reference point which was measured with the large range and less sensitive differential pressure transducer. As a result, the error of pressure measurements (5P) was determined by the accuracy of the large range transducer and was relatively large. The axial distributions of the measured data of transmural pressure, Plm, cross-sectional area, A, and fluid velocity, U=Q/ A, are shown in Figs. 7 and 8. The test-chamber pressures (Pe), the flow rates (Q, the estimated error of pressure measurements (8P) and a typical Reynolds number (Re) for each of the experiments are summarized in Table 2. The surfaces of the tubes in each of the steady flow experiments are also shown in Figs. 7 and 8. They were plotted by a commercial Journal of Biomechanical Engineering
30
:*
* t
Thick-Wall Tube H
i M t t t *
15
, 1
1
I < > I
x + + + +
Q = 122.9 ml/s (+) -4 800 * * * *
+ + + +
-
§ 400
Q = II9.2 mJ/s(*) "0.0
0.5
1.0
Fig. 7 Measurements of steady fluid flows through the thick-wall tube. (+) represent values for Q = 122.9 ml/s, (*) represent values for 0 = 119.2 ml/s. The dimensionless length is J = x/L, where L is the length of the measured region.
package that computes the surface from discrete data (e.g., contours of given cross-sections) and draws the surface with a predefined number of longitudinal and circumferential grid lines. The surfaces in Figs. 7 and 8 were generated from measured contours of eleven cross-sections along the measuring zone (L = 100 mm). In the present work, exact contours (about 35) were obtained at an axial resolution of 3 mm. This can be improved, if necessary, by increasing the number of raster lines projected on the measuring region. In some cases the tube was not symmetric about the horizontal plane and rotation of the cross-section can be observed (e.g., Fig. 8; Q = 38.8 ml/ FEBRUARY 1992, Vol. 114 / 89
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Thin-Wall Tube
x E iH.
Table 2 Reference parameters for the steady flows. Re = UDe/ v where De is the hydraulic diameter (4A/s), v is the kinematic viscosity, and s is the cross-sectional perimeter (assumed constant during collapse states)
Pe [cm H2OJ Q [ml/s] bP [cm H 2 0] Re(£ = 0) fT (Blasius)
'
Thin tube Case II Case I 59.03 57.04 43.2 38.8 0.285 0.295 2018 1790 0.012 0.012
Thick tube Case III Case IV 11.48 11.88 122.9 119.2 0.0594 0.0574 5,325 5165 0.009 0.009
-2-
E QT
1
