245
Effect of tube ovalling on pressure wave propagation speed Eur Ing A Anderson, BSc, PhD, CEng, MIMechE and G R Johnson, BSc, PhD Department of Mechanical, Materials and Manufacturing Engineering, University of Newcastle upon Tyne
For physiological and other flows it is often assumed that the pressure pulse wave speed is given by the classic Moens-Korteweg expression and this may be used, for example, to assist in the delermination of in vivo blood vessel wall incremental Young’s modulus. A number ojphysical factors affecting the value of this wave speed have been reviewed in the literature, but the effect of slight ovalling of the tube cross-section is rarely mentioned. The analysis for a tube ofelliptic cross-section shows that even a very small degree of ovalling can cause quite substantial reductions in Young mode waue propagation velocities compared with the classic Moens-Korteweg expression. Bending-induced changes in crosssection shape with internal pressure increase the apparent elast icily ofthe tube wall. Experimental confirmation is provided by waterhammer wave speed measurements in a copper tube that has been ovalled by coiling. Even though the Young mode is not dominant in this case, as it would be,for a physiological case, the measured wave speed is quite clearly less than the Moens-Korteweg theory and it can be shown that the small degree of measured tube niiality explains this. NOTATION
cross-sectional area of tube (normal to one-dimensional flow) major and minor radii of ellipse (Fig. 1) Helmholtz-Korteweg disturbance propagation wave velocity [equations ( 3 ) and (411 isothermal unconfined velocity of sound in liquid [equation (4)] Young mode (8) disturbance propagation wave velocity in distensible tube [equations (1) and (4)] diameter of circular cross-section tube or diameter of circular cross-section with same perimeter as elliptic cross-section tube [equation (25)] uniform wall thickness of thin-walled tube (e =‘3 4 complete elliptic integral of the second kind (1618) incremental Young’s modulus of elasticity for tube wall material hypergeometric function (1618) hoop and radial shear components of force per unit tube length acting on tube wall for internal gauge pressure p (Fig. 4) [equation ( I 6)] x and y components of force per unit tube length acting on tube wall for internal gauge pressure p (Fig. 3 and 4) tube wall moment of inertia per unit tube length [equation (21)J additional complete elliptic integrals [equations (14) and (23)] complete elliptic integral of the first kind (16-18) isothermal bulk modulus of elasticity for liquid of relatively low compressibility The M S was rrcrirred on 6 July 1990 and was accepted for puhlimtion on 23 October 1990.
H02690 Q IMechE 1990
index of tube cross-section ellipticity [equation (5)] bending moment per unit tube length due to tube ovality for internal gauge pressure P (Fig. 3) tube internal gauge pressure wall perimeter of elliptic cross-section tube [equation (1l)] distance around tube wall (Figs 1 and 2) time (independent variable) [equations (2) and (311 strain energy per unit tube length (total, due to hoop stresses, due to bending stresses) [equations (1 8), (21) and (24)] cross-section averagc one-dimensional flow velocity coordinates in plane of tube cross-section, normal to flow direction z (Fig. 1) distance along tube axis in onedimensional flow direction [equations (2) and (3)1 axial length of control volume [equation
(1)l
measure of influence of degree of ellipticity m on Young mode wave speed cy [equations (25) and (30)] density of liquid in tube hoop stress in tube wall [equation (17)] angles about ellipse centre (Fig. 1) angle of tangent to ellipse (Figs 1 and 2) 1 INTRODUCTION
Helmholtz is usually credited (1-4) as the first to suggest that the speed of pressure disturbance wave propagation in fluid-filled tubes ( c ) depends on the distensibility of both the fluid (that is unconfined speed of sound) and the tube wall. For the special case of liquids with relatively low compressibility contained in thin-walled elastic cylindrical tubes, the two different disciplines of hydraulic transients (5-7) and physiological flows (1, 8)
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A ANDERSON AND G R JOHNSON
246
present this in a familiar form first derived in its entirety by Korteweg (9):
where c, is the unconfined sonic velocity, with K , and p the fluid isothermal bulk modulus and density respectively and cy the ‘Young mode’ (8) disturbance propagation velocity with E, the wall material incremental Young’s modulus and wall thickness e 4 d (tube internal diameter). It should be noted that equation ( l ) ,and, henceforth, purely to minimize algebra for simplicity, assumes that the tube is of initially uniform crosssection and that wall radial and longitudinal strain effects can be ignored, the Poisson ratio corrections for these being well established in the literature [for example references (6),(7),(10) and (1111. Especially for physiological flows, Lambossy (1) and others [for example references (8) and (12)] have reviewed a range of such physical factors which can alter a measured wave propagation velocity (c) from the classic theoretical value of equation (1). The present objective is to illustrate one rarely mentioned effect that may be particularly common with in uiuo blood vessels but which may also occur with, for example, coiled tubes for heat exchangers, buried flexible piping etc., that is situations in which, initially, intentionally or assumed circular cross-section tubes may develop ovality. The effect of any ovality will be to increase the apparent elasticity of the pipe wall in equation (l), because changes in cross-sectional area will arise from not only hoop but also bending stresses that alter the crosssectional shape. This will be most significant where the ‘Young mode’ (cy) dominates, for example physiological flows, though it has been considered previously for nonphysiological straight-sided cross-sections (13, 14). The present objective is to evaluate the influence of tube ovality on wave speeds calculated by equation (1).
ing equation (3) with the corresponding momentum equation and rewriting these in characteristic form it can be demonstrated (4-7) that equation (4) defines the disturbance wave propagation velocity (c) and it is easy to confirm (4, 5) that equation (1) is the appropriate special case of this. 3 THEORY FOR TUBES OF ELLIPTIC CROSSECTION
‘Ovality’ can take many forms. Because it is conveniently characterized by a single parameter: (5)
(where a and b are the major and minor radii respectively) the elliptic cross-section (Fig. 1) is adopted as representative. A small increment in subtending angle (do) will give rise to a small element (ds) on the perimeter with components (dx, dy) parallel to the x and y coordinate axes (Fig. 1 ) : x=
+ a sin 8,
y = -bcos0, ds = J{(dx)2
dx = + a cos 0 d8 dy= +bsin8d8
+ (dy)’} = aJ(1
-m
sin2 0) d0
(6)
Considering equilibrium of a small element of the tube wall (Fig. 2) subject to internal gauge pressure p
’t I I
2 GENERAL EXPRESSION FOR WAVE SPEED
Expanding the conservation law form of the onedimensional flow continuity equation:
a
a
at
aZ
- @ A Az) + - @Au)AZ = 0 gives the non-conservation form:
aP aP au - + 0 - + pc2- = 0 at
(3)
aZ
aZ
in which the general expression for wave speed (4,5) is
-1 ---aP + -P- aA c2
ap
A ap
_1 - _aP - ap
:C
1 c;-
p aA A ap
(4)
where the independent variables are the axial space coordinate in the one-dimensional flow direction (2)and time ( t ) ; changes in fluid density @) and tube flow crosssectional area (A) are assumed to depend on fluid static pressure (p) only and u is fluid flow velocity. By combin-
Fig. 1 Ellipse geometry definition sketch Q IMechE 1990
Part H: Journal of Engineering in Medicine
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247
EFFECT OF TUBE OVALLING ON PRESSURE WAVE PROPAGATION SPEED
dF 4 F , + -2 ds ds
{ - b cos(8
v ,
+ do), a sin(8 + do)} \
pds
/M
+
f’
F,
+
dF ds
2 ds
Fig. 2 Equilibrium of elliptic tube wall element
then equilibrium (per unit length of tube) gives (15) dFX dB
-= + p a
dF d6
in which E(m) is the ‘complete elliptic integral of the second kind’ [see references (16) to (18) and others] :
cos 0,
J(l - m sin2 tl) dtl
4a =
- p h sin 8,
(7)
m
dM do
-= paZ - sin 8
2
From equations (8) to (10):
cos Q
so that at any angle 4 :
F,(qb) - Fx(0) =
5,”2
dB = p a sin qb
where the integral Z,(m) is defined as (18)
Il(m) =
By symmetry and equilibrium respectively: F,(O)
= P,
(3 -
(3
= 0 + F, - = pa,
F,(O)
= ph
0=
!
M ds
0
=!
MaJ(1
-
m sin2 Q) d8
0
where the perimeter P of the ellipse is defined by ds = 4aE(m)
(14)
and from equation (10):
rniz ’
sin2 OJ(1 - m sin2 tl) dtl
(9)
Similarly, symmetry requires that there is no change in the slope over a quadrant (P/4) after pressurization (E), that is using equation (6): f P/4
sb:’
(10)
Il(4 M(0) = - p -1 (a2 - b2) 2
The hoop (FH)and shearing (I;,) forces (Fig. 2) are defined by F , = F , cos $ - F, sin $, F,
= F,
F,
= F,
+ F, cos $ cos $ + F, sin I), sin $
Fs = -F, sin $
@ IMechE 1990
(16)
+ F, cos $ Proc Instn Mech Engrs Vol 204
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A ANDERSON AND G
248
R JOHNSON
For thin-walled tubes as in the conventional equation (1) the contribution of the radial shearing force I;, to strain energy can be assumed to be negligible and only hoop stresses (OH) need be considered. From equations (6) and (16): 1
OH --
/
2 ,o -
(1 7)
pb/e - J(l - m sin2 0)
/
K(m), equation (20), first kind E(m), equation (14). second kind
I,(rn), equation (14) f 2 ( m ) ,equation (23)
The hoop strain energy (V,) per unit length of tube is defined in the usual way (15): nI2 1
whence UHE, bb -- 2 - - K(m)
(19)
ae
p2a2
in which K(m) is the ‘complete elliptic integral of the first kind’ [see references (16) to (18) and others] 1
K(m) =
de
Unlike the conventional circular cross-section tube of equation (l), however, it is necessary to consider also the bending strain energy (UM)per unit length of tube (15):
whence W E p’a2
(!) {
I,(m) -
e
Values of the four complete elliptic integrals ( K , E , I , , I,) are shown in Fig. 3. Thus, the total strain energy per unit length in the tube wall (ignoring axial and radial effects) is, finally, UH
+ U,
=p
2
(nab)
=
4uE(m),
that is d
=a
and
0.7
I
0.75m
b
0.5 h/u
The use of energy methods for pressure surges has been reviewed by Wood and Stelson (19) but for the Joukowsky pressure rise rather than the wave speed as below. For no axial strain, then, per unit length of tube, volume change is given by area change (AA) and AU=pAA (27) whence, with equation (24) and ignoring for the moment the fluid compressibility (c, = cc),then equation (4) can be evaluated as p dU pd c1 pA dp E y e For the limiting case of a circular cross-section:
1
where d is the diameter of a circular cross-section with the same perimeter as the elliptic cross-section:
nd
I
0.64 0.6
4 WAVE PROPAGATION VELOCITY IN FLUID-FI [LED ELLIPTIC TUBE
sin4 dJ(1 - m sin’ 0) d0
=
I
0.51
Fig. 3 Values of complete elliptic integrals (16-18)
where the integral 12(m)is defined as (18)
U
1
0.36 0.8
0.9
1
3
= 6m2
I
0.19
4 E(m) n
-
(25)
pdA dp
-
cb- A
d a=b=2’
(28)
m=0,
and equation (28) reduces to the classic equation (1). In equation (28) the factor a, defined by equation (26), is the measure of the influence of the degree of ellipticity Part H Journ.4 of Engneenng In Medicine
0 IMechE 1990
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EFFECT O F TUBE OVALLING ON PRESSURE WAVE PROPAGATION SPEED
249
speed below the theoretical cylindrical value of equation (l),especially for thin-walled tubes. Other forms of ovalling will obviously have a similar general effect.
1
5 EXAMPLE
0.9
It is not easy to acquire a uniform elliptic cross-section tube in which the ‘Young mode’ wave speed is dominant to test the theory directly. However, even with waterhammer this effect may be sufficiently significant to be observable. A simple undergraduate teaching experimental apparatus for waterhammer uses a long copper tube coiled to save space. Regrettably, given its purpose, actual measured wave speeds do not correspond with the classic theory of equation (l),even when care is taken to eliminate all the common causes of this, for example dissolved air etc. Table 1 gives the values. The experimental wave speed (rows 4 and 5) can be determined from either wave frequency (f)or the amplitude of the Joukowsky pressure rise (Ap) for instantaneous flow stoppage from a velocity uo (6, 7):
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
I
I
1
I
0.19 0.9
0.36 0.8
0.51 0.7
0.64 0.6
I
0.75~1
0.5 b/cr
Degree of ellipticity
Fig. 4 Effect of ellipticity on wave speed, that is ratio of wave speed in elliptic tube to theoretical cylindrical tube value from equation (1)
(where L is the pipe length). The latter is less precise but confirms the central tendency of the former. The theoretical wave speeds (rows 1 to 3 ) are given from equation (1) with an appropriate correction (11) for Poisson ratio effects and the fact that d/e < 30 (this amounts only to a multiplier of 1.01 f 0.03 on the ‘Young mode’ c,; that is axial and radial effects are negligible as assume). Even at 95 per cent confidence limits for the uncertainties, it can be seen that there is no overlap between theory and experiment for the actual wave speed c (rows 3 and 4) or for the theoretical ‘Young mode’ c, (row 2) and the corresponding value deduced from the experiment (row 6) using
(m)on the ‘Young mode’ wave speed (c,):
Ratio of wave speed in ovalled tube [equation (28)] theoretical cylindrical tube wave speed [equation (l)] 1 - _ Ja
(30)
Typical values of this ratio are plotted in Fig. 4 and it can be seen that ellipticity (m > 0) can very substantially reduce the ‘Young mode’ (and hence overall) wave
The ratio from these experimental results is 0.7 f 0.1 with the more usual 65 per cent confidence limit (row 7). In checking the experimental procedures and apparatus data no cause for this discrepancy could be found other than that, though the copper tube had originally been within its specified tolerances, coiling it (even at a relatively large coil-tube diameter ratio) had caused the cross-section to oval slightly with b/a = 0.875 f 0.055 or m = 0.23 f 0.10 (65 per cent confidence limit). Even
Table 1 Examples of experimental wave speed less than theoretical value by equation (1) (95 per cent confidence limit) 1 2 3
Unconfined sonic speed c, ‘Young mode’ velocity c y Waterhammer wave speed c
1465 25 3400 300 1345 i 40
Experimental waterhammer wave speed c (m/s) [equation (31)]
4 5
Determined from frequency Determined from amplitude (check-not used below)
1250 & 40 1230 80
‘Young mode’ wave
6
Velocity cy (m/s) Ratio of actual/theoretical (a 1/2) [equation (30)]
2400 f 200 0.7 f 0 1 (65 per cent confidence limit)
Theoretical circular cross-section wave velocities (rnis) [equation (1)1
*
-~
velocity inferred from expenment [equation (32)]
7
*
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A ANDERSON AND G R JOHNSON
250
L
9.6 10.4
;1
Range of apparatus m values
I
I 1
I 0.2775 0.85
0. I9 0.9
0.0975 0.95
I 0.36m 0.8 b/n
Degree of ellipticity
Fig. 5 Comparison of theoretical and experimental effects of ellipticity on wave speed in water-filled copper tubing
though it is improbable that this ovalling is elliptic throughout, the theory [equations (26) and (30)] suggests that b/u = 0.85 f 0.05 or rn = 0.28 & 0.08 (Fig. 5), which overlaps significantly with the experimental values, confirming that this is indeed the probable cause of the discrepancy. At small degrees of ovality, the actual section shape is not likely to be very significant.
An experimental comparison has demonstrated this and also indicated that the analysis gwen for an elliptic tube will serve reasonably well in practice, even when there is no guarantee that the tube ovalling is necessarily of this particular form, providing that the equivalent degree of ellipticity is small. REFERENCES
6 CONCLUSIONS
As has been illustrated for an elliptic cross-section tube, whenever a tube cross-section deviates from the circular there may be significant reductions in the ‘Young mode’ wave velocity due to bending-induced changes in the tube cross-section (always striving to the ideal circular section with maximum area for a given perimeter). Figure 4 indicates that, even for quite small degrees of ellipticity, this effect may be considerably more significant than some of the others previously reported in the literature [for example references (I), (8), (12) etc.]. Attempts to, for example, determine the in viuo incremental Young’s modulus from pulse propagation velocities will overestimate this significantly if the vessel has any ovality.
1 Lambossy, P. A p e r p historique el critique sur le probleme de la propagation des ondes dans un liquide compressible enfermt dans un tube Blastique. Ilelv. Physiofogica et Pharmacoloyica Acta, 1950,8(2), 209-227; 1951, YZ), 145-161. 2 Wood, F. M. History of waterhammer. Civil Engineering research report 65, April 1970, Queens University, Kingston, Ontario. 3 Anderson, A. Menabrea’s note on waterhammer; 1858. ASCE Proc., 1.Hydraulics Diu., 1976, 102(HYl), 29-39. 4 Anderson, A. On the formulation of problems in hydraulic transients. PhD thesis, 1976, ch. 2, University of Aberdeen. 5 Raabe, J. Hydro power. T h e design, use and function of hydromechanical hydraulic and electrical equipment, 1985, Sec. 8.3 (VDI Verlag, Dusseldorf). 6 Evangelisti, G . Waterhammer analysis by the method of characteristics. L’Energia Elettrica, 1969, *lo), 673-692; 46(1 I), 759-771 ; 46(12), 839-858.’ 7 Wylie, E. B. and Streeter, V. L. Fluid transients, 1983 (FEB, Ann Arbor, Mich.).
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EFFECT OF TUBE OVALLING ON PRESSURE WAVE PKOPAGATION SPEED
8 McDonald, D. A. B l o o d j u w in arteries, 2nd edition, 1974, Ch. 10 (Edward Arnold, London). 9 Korteweg, D. J. Uber die Fortpflanzungsgeschwindigkeit des Schalles in elastischen Rohren. Annln der Physik und Chemie (Wiedemann),Neue Folge, 1878,5(12), 525-542. 10 Parmakian, J. Watrrhammer analysis, 1955, Chs 2 and 3 (PrenticeHall, New York). 11 Betamio de Almeida, A. Manual de proteccio contra o golpe de ariete em conduras eleualhrias, 1981, Sec. 8 (LNEC, Lisbon). 12 Nakoryakov V. E. et ai. Waterhammer and propagation of perturbations in elastic fluid-filled pipes. Fluid Dynamics (Consultants Bureau, New York), 1977,11,493498. 13 Thorley, A. R. D. and Guymer, C. Fundamental equations governing pressure surge phenomena in pipes of rectangular crosssection. Proc. Instn Mech. Engrs, 1975,189(40),325-332. 14 Thorley, A. R. D. and Twyman, J. W. R. Propagation of transient pressure waves in a sodium-cooled fast reactor. In Second Interna-
251
tional Conference on Pressure surges, London, September 1976, paper A1 (BHRA Fluid Engineering, Cranfield). 15 Timoshenko, S. Strength of materials. Part I . Elemmtury, 3rd edition, 1955, Chs 11 and 12 (van Nostrand Reinhold, New York). 16 Jahnke, E. and Emde, F. Funktionentafeln mit Formeln und Kuroen, 3rd edition, 1938 (Teubner, Leipzig and Berlin). 17 Abramowitz, M. et al. Pocketbook of mathematical functions, 1984 (Verlag Harri Deutsch, Frankfurt am Main). 18 Gradshteyn, I. S. and Ryzhik, I. M. Table of integrals, series and products (Ed. A. Jeffrey), 4th edition, 1965 (Academic Press, New York). 19 Wood, D. J. and Stelson, T. E. Energy analysis of pressure surges in closed conduits. In Deoelopments in theoretical and applied mechanics, Vol. 1, 1963, pp. 371-388 (Plenum Press, New York).
Downloaded from pih.sagepub.com at TU Muenchen on July 8, 2015
Effect of tube ovalling on pressure wave propagation speed Eur Ing A Anderson, BSc, PhD, CEng, MIMechE and G R Johnson, BSc, PhD Department of Mechanical, Materials and Manufacturing Engineering, University of Newcastle upon Tyne
For physiological and other flows it is often assumed that the pressure pulse wave speed is given by the classic Moens-Korteweg expression and this may be used, for example, to assist in the delermination of in vivo blood vessel wall incremental Young’s modulus. A number ojphysical factors affecting the value of this wave speed have been reviewed in the literature, but the effect of slight ovalling of the tube cross-section is rarely mentioned. The analysis for a tube ofelliptic cross-section shows that even a very small degree of ovalling can cause quite substantial reductions in Young mode waue propagation velocities compared with the classic Moens-Korteweg expression. Bending-induced changes in crosssection shape with internal pressure increase the apparent elast icily ofthe tube wall. Experimental confirmation is provided by waterhammer wave speed measurements in a copper tube that has been ovalled by coiling. Even though the Young mode is not dominant in this case, as it would be,for a physiological case, the measured wave speed is quite clearly less than the Moens-Korteweg theory and it can be shown that the small degree of measured tube niiality explains this. NOTATION
cross-sectional area of tube (normal to one-dimensional flow) major and minor radii of ellipse (Fig. 1) Helmholtz-Korteweg disturbance propagation wave velocity [equations ( 3 ) and (411 isothermal unconfined velocity of sound in liquid [equation (4)] Young mode (8) disturbance propagation wave velocity in distensible tube [equations (1) and (4)] diameter of circular cross-section tube or diameter of circular cross-section with same perimeter as elliptic cross-section tube [equation (25)] uniform wall thickness of thin-walled tube (e =‘3 4 complete elliptic integral of the second kind (1618) incremental Young’s modulus of elasticity for tube wall material hypergeometric function (1618) hoop and radial shear components of force per unit tube length acting on tube wall for internal gauge pressure p (Fig. 4) [equation ( I 6)] x and y components of force per unit tube length acting on tube wall for internal gauge pressure p (Fig. 3 and 4) tube wall moment of inertia per unit tube length [equation (21)J additional complete elliptic integrals [equations (14) and (23)] complete elliptic integral of the first kind (16-18) isothermal bulk modulus of elasticity for liquid of relatively low compressibility The M S was rrcrirred on 6 July 1990 and was accepted for puhlimtion on 23 October 1990.
H02690 Q IMechE 1990
index of tube cross-section ellipticity [equation (5)] bending moment per unit tube length due to tube ovality for internal gauge pressure P (Fig. 3) tube internal gauge pressure wall perimeter of elliptic cross-section tube [equation (1l)] distance around tube wall (Figs 1 and 2) time (independent variable) [equations (2) and (311 strain energy per unit tube length (total, due to hoop stresses, due to bending stresses) [equations (1 8), (21) and (24)] cross-section averagc one-dimensional flow velocity coordinates in plane of tube cross-section, normal to flow direction z (Fig. 1) distance along tube axis in onedimensional flow direction [equations (2) and (3)1 axial length of control volume [equation
(1)l
measure of influence of degree of ellipticity m on Young mode wave speed cy [equations (25) and (30)] density of liquid in tube hoop stress in tube wall [equation (17)] angles about ellipse centre (Fig. 1) angle of tangent to ellipse (Figs 1 and 2) 1 INTRODUCTION
Helmholtz is usually credited (1-4) as the first to suggest that the speed of pressure disturbance wave propagation in fluid-filled tubes ( c ) depends on the distensibility of both the fluid (that is unconfined speed of sound) and the tube wall. For the special case of liquids with relatively low compressibility contained in thin-walled elastic cylindrical tubes, the two different disciplines of hydraulic transients (5-7) and physiological flows (1, 8)
0954-41 19/90 $2.00 + .OS
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Proc Instn Mech Enprs Vol 204
A ANDERSON AND G R JOHNSON
246
present this in a familiar form first derived in its entirety by Korteweg (9):
where c, is the unconfined sonic velocity, with K , and p the fluid isothermal bulk modulus and density respectively and cy the ‘Young mode’ (8) disturbance propagation velocity with E, the wall material incremental Young’s modulus and wall thickness e 4 d (tube internal diameter). It should be noted that equation ( l ) ,and, henceforth, purely to minimize algebra for simplicity, assumes that the tube is of initially uniform crosssection and that wall radial and longitudinal strain effects can be ignored, the Poisson ratio corrections for these being well established in the literature [for example references (6),(7),(10) and (1111. Especially for physiological flows, Lambossy (1) and others [for example references (8) and (12)] have reviewed a range of such physical factors which can alter a measured wave propagation velocity (c) from the classic theoretical value of equation (1). The present objective is to illustrate one rarely mentioned effect that may be particularly common with in uiuo blood vessels but which may also occur with, for example, coiled tubes for heat exchangers, buried flexible piping etc., that is situations in which, initially, intentionally or assumed circular cross-section tubes may develop ovality. The effect of any ovality will be to increase the apparent elasticity of the pipe wall in equation (l), because changes in cross-sectional area will arise from not only hoop but also bending stresses that alter the crosssectional shape. This will be most significant where the ‘Young mode’ (cy) dominates, for example physiological flows, though it has been considered previously for nonphysiological straight-sided cross-sections (13, 14). The present objective is to evaluate the influence of tube ovality on wave speeds calculated by equation (1).
ing equation (3) with the corresponding momentum equation and rewriting these in characteristic form it can be demonstrated (4-7) that equation (4) defines the disturbance wave propagation velocity (c) and it is easy to confirm (4, 5) that equation (1) is the appropriate special case of this. 3 THEORY FOR TUBES OF ELLIPTIC CROSSECTION
‘Ovality’ can take many forms. Because it is conveniently characterized by a single parameter: (5)
(where a and b are the major and minor radii respectively) the elliptic cross-section (Fig. 1) is adopted as representative. A small increment in subtending angle (do) will give rise to a small element (ds) on the perimeter with components (dx, dy) parallel to the x and y coordinate axes (Fig. 1 ) : x=
+ a sin 8,
y = -bcos0, ds = J{(dx)2
dx = + a cos 0 d8 dy= +bsin8d8
+ (dy)’} = aJ(1
-m
sin2 0) d0
(6)
Considering equilibrium of a small element of the tube wall (Fig. 2) subject to internal gauge pressure p
’t I I
2 GENERAL EXPRESSION FOR WAVE SPEED
Expanding the conservation law form of the onedimensional flow continuity equation:
a
a
at
aZ
- @ A Az) + - @Au)AZ = 0 gives the non-conservation form:
aP aP au - + 0 - + pc2- = 0 at
(3)
aZ
aZ
in which the general expression for wave speed (4,5) is
-1 ---aP + -P- aA c2
ap
A ap
_1 - _aP - ap
:C
1 c;-
p aA A ap
(4)
where the independent variables are the axial space coordinate in the one-dimensional flow direction (2)and time ( t ) ; changes in fluid density @) and tube flow crosssectional area (A) are assumed to depend on fluid static pressure (p) only and u is fluid flow velocity. By combin-
Fig. 1 Ellipse geometry definition sketch Q IMechE 1990
Part H: Journal of Engineering in Medicine
Downloaded from pih.sagepub.com at TU Muenchen on July 8, 2015
247
EFFECT OF TUBE OVALLING ON PRESSURE WAVE PROPAGATION SPEED
dF 4 F , + -2 ds ds
{ - b cos(8
v ,
+ do), a sin(8 + do)} \
pds
/M
+
f’
F,
+
dF ds
2 ds
Fig. 2 Equilibrium of elliptic tube wall element
then equilibrium (per unit length of tube) gives (15) dFX dB
-= + p a
dF d6
in which E(m) is the ‘complete elliptic integral of the second kind’ [see references (16) to (18) and others] :
cos 0,
J(l - m sin2 tl) dtl
4a =
- p h sin 8,
(7)
m
dM do
-= paZ - sin 8
2
From equations (8) to (10):
cos Q
so that at any angle 4 :
F,(qb) - Fx(0) =
5,”2
dB = p a sin qb
where the integral Z,(m) is defined as (18)
Il(m) =
By symmetry and equilibrium respectively: F,(O)
= P,
(3 -
(3
= 0 + F, - = pa,
F,(O)
= ph
0=
!
M ds
0
=!
MaJ(1
-
m sin2 Q) d8
0
where the perimeter P of the ellipse is defined by ds = 4aE(m)
(14)
and from equation (10):
rniz ’
sin2 OJ(1 - m sin2 tl) dtl
(9)
Similarly, symmetry requires that there is no change in the slope over a quadrant (P/4) after pressurization (E), that is using equation (6): f P/4
sb:’
(10)
Il(4 M(0) = - p -1 (a2 - b2) 2
The hoop (FH)and shearing (I;,) forces (Fig. 2) are defined by F , = F , cos $ - F, sin $, F,
= F,
F,
= F,
+ F, cos $ cos $ + F, sin I), sin $
Fs = -F, sin $
@ IMechE 1990
(16)
+ F, cos $ Proc Instn Mech Engrs Vol 204
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A ANDERSON AND G
248
R JOHNSON
For thin-walled tubes as in the conventional equation (1) the contribution of the radial shearing force I;, to strain energy can be assumed to be negligible and only hoop stresses (OH) need be considered. From equations (6) and (16): 1
OH --
/
2 ,o -
(1 7)
pb/e - J(l - m sin2 0)
/
K(m), equation (20), first kind E(m), equation (14). second kind
I,(rn), equation (14) f 2 ( m ) ,equation (23)
The hoop strain energy (V,) per unit length of tube is defined in the usual way (15): nI2 1
whence UHE, bb -- 2 - - K(m)
(19)
ae
p2a2
in which K(m) is the ‘complete elliptic integral of the first kind’ [see references (16) to (18) and others] 1
K(m) =
de
Unlike the conventional circular cross-section tube of equation (l), however, it is necessary to consider also the bending strain energy (UM)per unit length of tube (15):
whence W E p’a2
(!) {
I,(m) -
e
Values of the four complete elliptic integrals ( K , E , I , , I,) are shown in Fig. 3. Thus, the total strain energy per unit length in the tube wall (ignoring axial and radial effects) is, finally, UH
+ U,
=p
2
(nab)
=
4uE(m),
that is d
=a
and
0.7
I
0.75m
b
0.5 h/u
The use of energy methods for pressure surges has been reviewed by Wood and Stelson (19) but for the Joukowsky pressure rise rather than the wave speed as below. For no axial strain, then, per unit length of tube, volume change is given by area change (AA) and AU=pAA (27) whence, with equation (24) and ignoring for the moment the fluid compressibility (c, = cc),then equation (4) can be evaluated as p dU pd c1 pA dp E y e For the limiting case of a circular cross-section:
1
where d is the diameter of a circular cross-section with the same perimeter as the elliptic cross-section:
nd
I
0.64 0.6
4 WAVE PROPAGATION VELOCITY IN FLUID-FI [LED ELLIPTIC TUBE
sin4 dJ(1 - m sin’ 0) d0
=
I
0.51
Fig. 3 Values of complete elliptic integrals (16-18)
where the integral 12(m)is defined as (18)
U
1
0.36 0.8
0.9
1
3
= 6m2
I
0.19
4 E(m) n
-
(25)
pdA dp
-
cb- A
d a=b=2’
(28)
m=0,
and equation (28) reduces to the classic equation (1). In equation (28) the factor a, defined by equation (26), is the measure of the influence of the degree of ellipticity Part H Journ.4 of Engneenng In Medicine
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EFFECT O F TUBE OVALLING ON PRESSURE WAVE PROPAGATION SPEED
249
speed below the theoretical cylindrical value of equation (l),especially for thin-walled tubes. Other forms of ovalling will obviously have a similar general effect.
1
5 EXAMPLE
0.9
It is not easy to acquire a uniform elliptic cross-section tube in which the ‘Young mode’ wave speed is dominant to test the theory directly. However, even with waterhammer this effect may be sufficiently significant to be observable. A simple undergraduate teaching experimental apparatus for waterhammer uses a long copper tube coiled to save space. Regrettably, given its purpose, actual measured wave speeds do not correspond with the classic theory of equation (l),even when care is taken to eliminate all the common causes of this, for example dissolved air etc. Table 1 gives the values. The experimental wave speed (rows 4 and 5) can be determined from either wave frequency (f)or the amplitude of the Joukowsky pressure rise (Ap) for instantaneous flow stoppage from a velocity uo (6, 7):
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
I
I
1
I
0.19 0.9
0.36 0.8
0.51 0.7
0.64 0.6
I
0.75~1
0.5 b/cr
Degree of ellipticity
Fig. 4 Effect of ellipticity on wave speed, that is ratio of wave speed in elliptic tube to theoretical cylindrical tube value from equation (1)
(where L is the pipe length). The latter is less precise but confirms the central tendency of the former. The theoretical wave speeds (rows 1 to 3 ) are given from equation (1) with an appropriate correction (11) for Poisson ratio effects and the fact that d/e < 30 (this amounts only to a multiplier of 1.01 f 0.03 on the ‘Young mode’ c,; that is axial and radial effects are negligible as assume). Even at 95 per cent confidence limits for the uncertainties, it can be seen that there is no overlap between theory and experiment for the actual wave speed c (rows 3 and 4) or for the theoretical ‘Young mode’ c, (row 2) and the corresponding value deduced from the experiment (row 6) using
(m)on the ‘Young mode’ wave speed (c,):
Ratio of wave speed in ovalled tube [equation (28)] theoretical cylindrical tube wave speed [equation (l)] 1 - _ Ja
(30)
Typical values of this ratio are plotted in Fig. 4 and it can be seen that ellipticity (m > 0) can very substantially reduce the ‘Young mode’ (and hence overall) wave
The ratio from these experimental results is 0.7 f 0.1 with the more usual 65 per cent confidence limit (row 7). In checking the experimental procedures and apparatus data no cause for this discrepancy could be found other than that, though the copper tube had originally been within its specified tolerances, coiling it (even at a relatively large coil-tube diameter ratio) had caused the cross-section to oval slightly with b/a = 0.875 f 0.055 or m = 0.23 f 0.10 (65 per cent confidence limit). Even
Table 1 Examples of experimental wave speed less than theoretical value by equation (1) (95 per cent confidence limit) 1 2 3
Unconfined sonic speed c, ‘Young mode’ velocity c y Waterhammer wave speed c
1465 25 3400 300 1345 i 40
Experimental waterhammer wave speed c (m/s) [equation (31)]
4 5
Determined from frequency Determined from amplitude (check-not used below)
1250 & 40 1230 80
‘Young mode’ wave
6
Velocity cy (m/s) Ratio of actual/theoretical (a 1/2) [equation (30)]
2400 f 200 0.7 f 0 1 (65 per cent confidence limit)
Theoretical circular cross-section wave velocities (rnis) [equation (1)1
*
-~
velocity inferred from expenment [equation (32)]
7
*
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A ANDERSON AND G R JOHNSON
250
L
9.6 10.4
;1
Range of apparatus m values
I
I 1
I 0.2775 0.85
0. I9 0.9
0.0975 0.95
I 0.36m 0.8 b/n
Degree of ellipticity
Fig. 5 Comparison of theoretical and experimental effects of ellipticity on wave speed in water-filled copper tubing
though it is improbable that this ovalling is elliptic throughout, the theory [equations (26) and (30)] suggests that b/u = 0.85 f 0.05 or rn = 0.28 & 0.08 (Fig. 5), which overlaps significantly with the experimental values, confirming that this is indeed the probable cause of the discrepancy. At small degrees of ovality, the actual section shape is not likely to be very significant.
An experimental comparison has demonstrated this and also indicated that the analysis gwen for an elliptic tube will serve reasonably well in practice, even when there is no guarantee that the tube ovalling is necessarily of this particular form, providing that the equivalent degree of ellipticity is small. REFERENCES
6 CONCLUSIONS
As has been illustrated for an elliptic cross-section tube, whenever a tube cross-section deviates from the circular there may be significant reductions in the ‘Young mode’ wave velocity due to bending-induced changes in the tube cross-section (always striving to the ideal circular section with maximum area for a given perimeter). Figure 4 indicates that, even for quite small degrees of ellipticity, this effect may be considerably more significant than some of the others previously reported in the literature [for example references (I), (8), (12) etc.]. Attempts to, for example, determine the in viuo incremental Young’s modulus from pulse propagation velocities will overestimate this significantly if the vessel has any ovality.
1 Lambossy, P. A p e r p historique el critique sur le probleme de la propagation des ondes dans un liquide compressible enfermt dans un tube Blastique. Ilelv. Physiofogica et Pharmacoloyica Acta, 1950,8(2), 209-227; 1951, YZ), 145-161. 2 Wood, F. M. History of waterhammer. Civil Engineering research report 65, April 1970, Queens University, Kingston, Ontario. 3 Anderson, A. Menabrea’s note on waterhammer; 1858. ASCE Proc., 1.Hydraulics Diu., 1976, 102(HYl), 29-39. 4 Anderson, A. On the formulation of problems in hydraulic transients. PhD thesis, 1976, ch. 2, University of Aberdeen. 5 Raabe, J. Hydro power. T h e design, use and function of hydromechanical hydraulic and electrical equipment, 1985, Sec. 8.3 (VDI Verlag, Dusseldorf). 6 Evangelisti, G . Waterhammer analysis by the method of characteristics. L’Energia Elettrica, 1969, *lo), 673-692; 46(1 I), 759-771 ; 46(12), 839-858.’ 7 Wylie, E. B. and Streeter, V. L. Fluid transients, 1983 (FEB, Ann Arbor, Mich.).
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EFFECT OF TUBE OVALLING ON PRESSURE WAVE PKOPAGATION SPEED
8 McDonald, D. A. B l o o d j u w in arteries, 2nd edition, 1974, Ch. 10 (Edward Arnold, London). 9 Korteweg, D. J. Uber die Fortpflanzungsgeschwindigkeit des Schalles in elastischen Rohren. Annln der Physik und Chemie (Wiedemann),Neue Folge, 1878,5(12), 525-542. 10 Parmakian, J. Watrrhammer analysis, 1955, Chs 2 and 3 (PrenticeHall, New York). 11 Betamio de Almeida, A. Manual de proteccio contra o golpe de ariete em conduras eleualhrias, 1981, Sec. 8 (LNEC, Lisbon). 12 Nakoryakov V. E. et ai. Waterhammer and propagation of perturbations in elastic fluid-filled pipes. Fluid Dynamics (Consultants Bureau, New York), 1977,11,493498. 13 Thorley, A. R. D. and Guymer, C. Fundamental equations governing pressure surge phenomena in pipes of rectangular crosssection. Proc. Instn Mech. Engrs, 1975,189(40),325-332. 14 Thorley, A. R. D. and Twyman, J. W. R. Propagation of transient pressure waves in a sodium-cooled fast reactor. In Second Interna-
251
tional Conference on Pressure surges, London, September 1976, paper A1 (BHRA Fluid Engineering, Cranfield). 15 Timoshenko, S. Strength of materials. Part I . Elemmtury, 3rd edition, 1955, Chs 11 and 12 (van Nostrand Reinhold, New York). 16 Jahnke, E. and Emde, F. Funktionentafeln mit Formeln und Kuroen, 3rd edition, 1938 (Teubner, Leipzig and Berlin). 17 Abramowitz, M. et al. Pocketbook of mathematical functions, 1984 (Verlag Harri Deutsch, Frankfurt am Main). 18 Gradshteyn, I. S. and Ryzhik, I. M. Table of integrals, series and products (Ed. A. Jeffrey), 4th edition, 1965 (Academic Press, New York). 19 Wood, D. J. and Stelson, T. E. Energy analysis of pressure surges in closed conduits. In Deoelopments in theoretical and applied mechanics, Vol. 1, 1963, pp. 371-388 (Plenum Press, New York).
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