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Slurry transport in horizontal vertical pipelines is one of the major means of transport of ss gravels in the dredging industry. There exist 4 main flow regimes, the fixed or stationary bed regime, the sliding bed regime, the heterogeneous flow regime the homogeneous flow regime. Of course the transitions between the regimes are not very sharp, depending on parameters like the particle size distribution. The focus in this paper is on the homogeneous regime. Often the so called equivalent liquid model (ELM) is applied, however many researchers found hydraulic gradients smaller than predicted with the ELM, but larger that the hydraulic gradient of liquid. Talmon (2011, 2013) derived a fundamental equation (method) proving that the hydraulic gradient can be smaller than predicted by the ELM, based on the assumption of a particle free viscous sub-layer. He used a 2D velocity distribution without a concentration distribution. In this paper 5 methods are described ( derived) to determine the hydraulic gradient in homogeneous flow, of which the last method is based on pipe flow with a concentration distribution. It appears that the use of von Driest (Schlichting, 1968) damping, if present, dominates the results, however applying a concentration distribution may neutralise this. The final equation contains both the damping a concentration distribution giving the possibility to calibrate the constant in the equation with experimental data. The final equation is flexible gives a good match with experimental results in vertical horizontal pipelines for a value of ACv = 1.3. Data of horizontal experiments Dp = 0.050.30 m, d = 0.04 mm, vertical experiments Dp = 0.026 m, d = 0.125, 0.345, 0.560, 0.750 mm. Keywords: Slurry transport; Homogeneous transport; Viscous sub layer; Mixing length. INTRODUCTION Slurry transport in horizontal vertical pipelines is one of the major means of transport of ss gravels in the dredging industry. There exist 4 main flow regimes, the fixed or stationary bed regime, the sliding bed regime, the heterogeneous flow regime the homogeneous flow regime. Of course the transitions between the regimes are not very sharp, depending on parameters like the particle size distribution. In the case of very fine particles /or very high line speeds, the mixture is often considered to be a liquid with the liquid density l equal to the mixture density m, where the liquid density l can be replaced by the mixture density m in the hydraulic gradient equations. The velocity profile in a cross section of the pipe is considered to be symmetrical the slip between the particles the liquid is considered negligible. The concentration is assumed to be uniform over the cross section. Thus, the transport (or delivered) concentration the spatial concentration are almost equal will be named Cv. This is often referred to as the equivalent liquid model (ELM). Since the pressure losses are often expressed in terms of the hydraulic gradient, first some basic equations for the hydraulic gradient the relative excess hydraulic gradient (solids effect) are given. The hydraulic gradient according to the equivalent liquid model is: where it is assumed that the Darcy-Weisbach friction factors for liquid l mixture m are equal. This can also be written as: im = il (1 + Rsd Cv ) (2) Newitt et al. (1955) found that only 60% of the solids weight should contribute to the mixture density in order to obtain the equivalent liquid model, but this depends on the line speed possibly on other parameters as well. Many others also found hydraulic gradients below the ELM at high line speeds. Wilson et al. (1992) explain this with the effect of near wall lift resulting in an almost particle free viscous sub layer. However for very small particles values are found giving a higher value of the pressure gradient, which is often explained by correcting (increase) the apparent kinematic viscosity, for example with the Thomas (1965) equation. The pressure losses can be shown in an almost dimensionless form in a double logarithmic graph with the relative excess hydraulic gradient Erhg as the ordinate the hydraulic liquid gradient il as the abscissa. In terms of the relative excess hydraulic gradient, Erhg, the above equation can be written as: Erhg = im - il v2 = l ls = il Rsd Cv 2 g D p (3) im = pm v2 = m ls m l g L 2 g Dp l (1) m = l So in the Erhg(il) graph the above equation results in a straight line giving Erhg = il. Fig. 1 shows experimental data of Thomas (1976) of d = 0.04 mm iron ore in a Dp = 0.1585 m horizontal pipe versus the Delft Head Loss & Limit Deposit Velocity (DHLLDV) model, where the 4 term Thomas (1965) viscosity equation the homogeneous flow correction equation (37) with ACv = 1.3 are implemented. The match is remarkable. Relative excess hydraulic gradient Erhg vs. Hydraulic gradient il Fixed Bed Cvs=c. Sliding Bed Cvs=c. Mean Heterogeneous Flow Cvs=c. Homogeneous Flow Cvs=Cvt=c. 0.100 Resulting Erhg curve Cvs=c. Fixed Bed, Sliding Bed & Het. Flow Cvt=c. Fixed Bed, Sliding Bed & Sliding Flow Cvt=c. 0.010 Limit Deposit Velocity Ratio Potential/Kinetic Energy Cv=0.240 0.001 0.001 0.010 0.100 1.000 Relative excess hydraulic gradient Erhg (-) Hydraulic gradient il (-) © S.A.M. Dp=0.1585 m, d=0.04 mm, Rsd=4.00, Cv=0.240, sf=0.415 Fig. 1. The Thomas (1976) experimental data in a Erhg(il) graph. APPROACH In order to test the Talmon (2011, 2013) method to check if there are alternative methods the following approach is followed: 1. Method 1: First the Talmon (2011, 2013) method is discussed briefly. 2. Method 2: Since the Talmon (2011, 2013) method uses a 2D approach, with von Driest damping (Schlichting, 1968), but without a real concentration profile, in this second method the equations are derived for pipe flow with the Nikuradse (1933) mixing length equation, without von Driest damping (Schlichting, 1968) without a real concentration profile. The results are corrected for the volume flow. 3. Method 3: Von Driest damping (Schlichting, 1968) is added to method 2, resulting in a velocity profile comparable very close to the Talmon (2011, 2013) method 1. So method 1 method 3 are equivalent. 4. Method 4: The "Law of the Wall" 2D approach without von Driest damping (Schlichting, 1968). 5. The 4 methods are compared an equation describing the average behavior is derived. 6. Method 5: Finally a concentration profile is added to method 2. This method can simulate all previous methods, depending on the concentration profile chosen. 7. Based on experiments a value for the parameters of the concentration profile is chosen. METHOD 1: THE TALMON (2011) & (2013) HOMOGENEOUS REGIME EQUATION Talmon (2011, 2013) derived an equation to correct the homogeneous equation (the ELM model) for the slurry density, based on the hypothesis that the viscous sub-layer hardly contains solids at very high line speeds in the homogeneous regime. This theory results in a reduction of the resistance compared with the ELM, but the resistance is still higher than the resistance of clear liquid. Talmon (2011, 2013) used the Prtl approach for the mixing length, which is a 2D approach for open channel flow with a free surface. The Prtl approach was extended with damping near the wall to take into account the viscous effects near the wall, according to von Driest (Schlichting, 1968). The Talmon (2011, 2013) approach resulted in the following equation, with h = 6.7: m 1 = 2 l h l Rsd Cv + 1 Erhg Rsd Cv + 1 - h l Rsd Cv + 1 8 = i = il E l 2 l Rsd Cv + 1 Rsd Cv h 8 (4) This equation underestimates the hydraulic gradient (overestimates the effect of a particle free viscous sub layer) in a number of cases (small large particles) as Talmon (2011, 2013) proves with the examples shown in his papers. Only for d50 = 0.37 mm Dp = 0.15 m (medium particles) there is a good match. The philosophy behind this theory, combining a viscous sub-layer with liquid with a kernel with mixture, is however very interesting, because it explains fundamentally why the hydraulic gradient can be lower than the hydraulic gradient according to the ELM, as has been shown by many researchers. The model has been derived using the stard mixing length equation for 2D flow without a concentration distribution. When reproducing this method it was found that the coefficient h is not a constant but this coefficient depends on the value of Rsd·Cv according to Fig. 2. The value of 6.7 is found for a value of about 0.6 of the abscissa. METHOD 2: THE APPROACH USING THE NIKURADSE (1933) MIXING LENGTH The concept of Talmon (2011, 2013) is adopted, but modified by using pipe flow with the Prtl (1925) Nikuradse (1933) Homogeneous factor h (-) vs. Rsd·Cv Homogeneous factor h (-) y = 0.0098x4 - 0.1223x3 + 0.6187x2 - 1.9302x + 7.7426 R² = 1 © S.A.M. Rsd·Cv (-) Fig. 2. The coefficient h as a function of Rsd·Cv for method 1. mixing length equations, a linear shear stress distribution with a maximum at the pipe wall zero in the center a concentration distribution, assuming that in the homogeneous regime the mixture can be considered a Newtonian liquid with properties slightly different from those of water. The shear stress between the mixture, the slurry, the pipe wall is the sum of the viscous shear stress the turbulent shear stress: A required condition for pipe flow is, that the integral of the velocity over the pipe cross-section equals the average line speed times the cross-section, so: z 2 ( R - z ) = vls R (8) = + t = u u + t z z u u = m m + m t z z u t = 2 z The Nikuradse (1933) equation for the mixing length in pipe flow for large Reynolds numbers is: (5) z z = 0.14 - 0.08 1 - - 0.06 1 - R R R 2 4 4 3 z z = 0.14 1 - 1 - - 1 - 7 R 7 R (9) Now the shear stress can be expressed as (with z the distance from the pipe wall): = m ( ) = m m R = D p / 2 R-z R u u + 2 z z (6) This is a second degree function of the velocity gradient. Solving this with respect to the velocity gradient gives: u = z m 2 R-z + m + 4 2 ( ) m R m 2 2 2 ( ) The velocity profile can be determined by numerical integration. This velocity profile however, should result in an average velocity equal to the line speed used to determine the friction velocity. This appeared to be valid for very high line speeds (Reynolds numbers) in the range of 13001500 m/s, which is far above the range dredging companies operate (310 m/s). For line speeds in the range 310 m/s, the velocity profile resulted in an average velocity smaller than the line speed with a facto.81.0. Introducing a factor an extra term in the mixing length equation solves this problem. The original Nikuradse (1933) equation is multiplied with a factor an extra term is added to ensure that /R = ·z for z = 0, like is the case in the original equation. The factor is determined for each calculation in such a way, that the flow following from the line speed times the cross section of the pipe, equals the flow from integration of the velocity profile. 2 z 0.14 - 0.08 1 - R = 4 R -0.06 1 - z R (7) m 2 R-z + m + 4 2 ( ) m R m (10) + (1 - ) z - 0.00002 R e R Velocity distribution pipe 1 m diameter Normalized velocity (-) vls=1 m/sec vls=10 m/sec vls=100 m/sec vls=1000 m/sec z coordinate (m) © S.A.M. Fig. 3. The resulting velocity distributions in a Dp = 1 m pipe, corrected for the flow. Fig. 3 shows the resulting velocity distributions for a 1 m diameter pipe. Now the concept is, that a mixture flow with liquid as a carrier liquid in the viscous sub layer will have a lower resistance than a mixture flow with mixture in the viscous sub-layer. One can also say that in order to get the same pipeline resistance, the velocity in the center of the pipe of mixture with liquid in the viscous sub-layer um has to be higher than the case with mixture or liquid in the whole pipe ul. Assuming that the dynamic viscosity of the mixture is equal to the dynamic viscosity of the carrier liquid, m = l, in the viscous sub-layer the boundary layer where no solids are present, gives: 2 R-z - l l + l l + 4 2 ( ) m m R 2 2 difference by the average liquid velocity ul or vls,l results in a factor F, which only depends on the volumetric concentration Cv, the relative submerged density Rsd slightly on the line speed vls in the range 310 m/s on the pipe diameter Dp through the Darcy-Weisbach friction factor l, according to: F= um - ul vls,m - vls,l = = h l Rsd Cv ul vls,l = h l m -1 l (13) The shear stress at the pipe wall of a Newtonian liquid is by definition: u = z (11) l ( ) = 2 l vls,l Now assuming that the term with the density ratio is relevant only near the pipe wall not in the center of the pipe, this equation will simulate a mixture with liquid in the viscous sublayer. In fact, the density ratio reduces the effect of the kinematic viscosity, which mainly affects the viscous sub-layer. The velocity difference in the center of the pipe between mixture liquid, um ul, can now be determined (14) m ( ) = 2 m vls,m u u um - ul = - z m z l 0 0 R- 0 From this a relation for the ratio of the Darcy-Weisbach friction coefficients of a flow with mixture in the center carrier liquid in the viscous sub-layer to a flow with 100% liquid can be derived. 2 vls,l = 2 vls,m 2 l 2 R-z + l l + 4 2 ( ) m l m R 2 2 - 0 R - l (12) m vls,l = 2 l vls,m vls,l m 1 = = 2 l ( F v + v ) ( F + 1)2 ls,l ls,l (15) ( l )2 + 4 2 ( )2 R-z R This velocity difference, in the center of the pipe, is about equal to the difference of the average line speeds, however both can be determined numerically. Further it appears from the numerical solution of this equation, that dividing the velocity Equation (15) is independent of the method used, but the factor F is. Substituting the factor F from equation (13) gives: m 1 1 = = l ( F + 1)2 (h l Rsd Cv + 1)2 (16) This ratio depends on the homogeneous factor h, the Darcy-Weisbach friction factor l, the volumetric concentration Cv the relative submerged density Rsd. The ratio of the hydraulic gradients is now: im m m 1 + Rsd Cv = = il l l ( h l Rsd Cv + 1)2 im = il (h l Rsd Cv + 1)2 1 + Rsd Cv (17) This gives for the excess hydraulic gradient im il (the solids effect): im - il = il ( h l Rsd Cv + 1)2 1 + Rsd Cv ( R C + 1)2 - il h l sd v ( h l Rsd Cv + 1)2 2 1 + Rsd Cv - ( h l Rsd Cv + 1) = il ( h l Rsd Cv + 1)2 The relative excess hydraulic gradient Erhg is now: (18) For s gravel with a density of 2.65 ton/m3, the factor h is about 9.3, almost independent of the pipe diameter Dp the line speed vls for pipes with diameters of 0.5 m up to 1.2 m line speeds from 2 m/s up to 10 m/s. For very small pipes very low line speeds, like Dp = 0.1 m vls = 1 m/s, this factor decreases to about 8.5. The factor h is not 100% linear with the term Rsd·Cv for ss with a density of 2.65 ton/m3 volumetric concentrations up to 3540%. Since the solution depends on Rsd·Cv combined, the factor h also depends on this not on Rsd Cv separately. Fig. 4 shows the dependency of the factor h on the relative excess density Rsd·Cv. The factor E decreases with increasing concentration relative submerged density of the solids increases with increasing line speed. At normal line speeds (36 m/s) concentrations (0.10.3) this factor is about 0.740.78 (see Fig. 9). A larger pipe gives less reduction. This is caused by the smaller DarcyWeisbach friction coefficient l of larger pipes. METHOD 3: ADDING THE VON DRIEST DAMPING TO METHOD 2 Talmon (2013) used the Prtl approach for the mixing length, which is a 2D approach for open channel flow with a free surface. The Prtl approach was extended with damping near the wall to take into account the viscous effects near the wall, according to von Driest (Schlichting, 1968): E rhg = im - il Rsd C v 1 + Rsd C v - ( h l Rsd C v + 1) Rsd Cv ( h l Rsd C v + 1) Prl : (19) = z = il von Driest : = z 1 - e - z with : z+ = z /A (21) = E il The limiting value for the relative excess hydraulic gradient Erhg for a volumetric concentration Cv approaching zero, becomes: fl A=26 Erhg = im - il = il (1 - 2 h l ) Rsd Cv (20) Fig. 7 Fig. 5 show the velocity profile the mixing length profile of the Talmon (2013) approach with von Driest (Schlichting, 1968) damping the Nikuradse (1933) approach without damping. In both cases, the mixing length equations have been corrected in order to get the correct volume flow. There is a clear difference of the velocity profiles. Homogeneous factor h (-) vs. Rsd·Cv Homogeneous factor h (-) y = 0.0331x4 - 0.4023x3 + 1.9271x2 - 5.141x + 11.391 R² = 0.9999 © S.A.M. Rsd·Cv (-) Fig. 4. The homogeneous factor h as a function of Rsd·Cv for method 2. Mixing length vs. Distance to the wall z Mixing length (m) von Driest Nikuradse Prl © S.A.M. Distance from the wall z (m) Fig. 5. The mixing length versus the distance to the wall for a Dp = 1 m pipe at a line speed vls = 5 m/s, Prtl = 0.4, vonDriest = 0.3915, Nikuradse = 0.4, without von Driest (Schlichting, 1968) damping. Mixing length vs. Distance to the wall z Mixing length (m) von Driest Nikuradse Prl © S.A.M. Distance from the wall z (m) Fig. 6. The mixing length versus the distance to the wall for a Dp = 1 m pipe at a line speed vls = 5 m/s, Prtl = 0.4, vonDriest = 0.3915, Nikuradse = 0.4, with von Driest (Schlichting, 1968) damping. Applying the von Driest (Schlichting, 1968) damping to the Nikuradse (1933) equation (9) for the mixing length in pipe flow for large Reynolds numbers according to: 2 4 4 z 3 z = 0.14 1 - 1 - - 1 - 7 7 R R R 1- e -z+ / A (22) wall, dominates the effect of a particle free viscous sub layer, as expected. The results are almost independent of the pipe diameter Dp the line speed vls. Fig. 8 shows that the velocity profiles determined with equation (21) (Talmon) equation (22) (Miedema) are almost the same the behavior with respect to the hydraulic gradient reduction is equivalent. METHOD 4: THE LAW OF THE WALL APPROACH Gives almost exactly the same results as the Talmon (2013) approach, although the mixing length is completely different as is shown in Fig. 6. Only very close to the wall, where the viscous effects dominate, the same mixing lengths are found. Apparently, the von Driest damping, effective close to the pipe Often for open channel flow the so called "Law of the Wall" equations are used. Since in dredging the pipe wall is assumed to be smooth due to the continuous sing of the pipe wall, the smooth wall approach is discussed here. Based on the following assumption for the mixing length by Prtl (1925) the assumption that the viscous shear stress is negligible in the turbulent region, the famous logarithmic velocity equation, "Law of the Wall" for the turbulent flow is derived: The velocity difference at the center of the pipe is now: um ( R ) - ul ( R ) = ln m l R R u = * ln - ln z0,m z0,l = R m R ln - ln 0.11 l l 0.11 l z du = wall 1 - = l 2 R z = z 1 - R (23) (27) This "Law of the Wall" is also a 2D approach for open channel flow does not correct for pipe flow. The general equation for the velocity profile as a function of the distance to the smooth wall is: This gives for the Darcy-Weisbach friction coefficient ratio: ul m vls,l ul2 = 2 = 2 = 2 l vls,m um u * ln m + ul l = 1 1 m l + 1 ln l 8 = z u ( z ) = ln z0 z0 = 0.11 (24) (28) For the 100% liquid (or mixture) the velocity profile is defined as: z ul ( z ) = ln z0,l ul z0,l = 0.11 (25) The relative excess hydraulic gradient Erhg is now: Erhg = im - il = E il Rsd Cv For the mixture with liquid in the viscous sub-layer the velocity profile can be defined as: z um ( z ) = ln z0,m z0,m = 0.11 l l m (26) 1 1 + Rsd Cv - ln m l + 1 l 8 = il 2 1 Rsd Cv ln m l + 1 l 8 (29) Velocity ul, um vs. Distance to the wall z Velocity ul, um (m/sec) Talmon (liquid) Miedema (liquid) Law of the wall (liquid) Talmon (mixture) Miedema (mixture) Law of the wall (mixture) © S.A.M. Distance to the wall z (m) Fig. 7. The velocity versus the distance to the wall for a Dp = 1 m pipe at a line speed vls = 5 m/s, Prtl = 0.4, vonDriest = 0.3915, Nikuradse = 0.4, without von Driest (Schlichting, 1968) damping. Velocity ul, um vs. Distance to the wall z Velocity ul, um (m/sec) Talmon (liquid) Miedema (liquid) Law of the wall (liquid) Talmon (mixture) Miedema (mixture) Law of the wall (mixture) © S.A.M. Distance to the wall z (m) Fig. 8. The velocity versus the distance to the wall for a Dp = 1 m pipe at a line speed vls = 5 m/s, Prtl = 0.4, vonDriest = 0.3915, Nikuradse = 0.4, with von Driest (Schlichting, 1968) damping. COMPARISON OF THE MODELS Now 3 formulations are found for the reduction of the DarcyWeisbach friction factor the relative excess hydraulic gradient Erhg for slurry transport of a mixture with pure carrier liquid (water) in the viscous sub-layer, these are equations (4), (16) & (19) (28) & (29): Law of the Wall (no damping) Miedema (28) m 1 = 2 l 1 ln (1 + Rsd Cv ) l + 1 8 = 0.4 Nikuradse (no damping) Miedema (16) m 1 = l ( h l Rsd Cv + 1)2 h = 9.3 Prl - von Driest (damping) Talmon (4) m 1 = 2 l h l Rsd Cv + 1 8 h = 6.7 friction factor ratios the relative excess pressure gradient coefficient E. The methods 2 4 without damping do not differ too much, both give a reduction on the solids effect of about 1826% in s for medium concentrations. The methods 1 3 with damping however give a reduction on the solids effect of about 5565% in s, almost 3 times as much. For the virtual solid with a solids density of 10 ton/m3, the reductions are 1830% 6580%. Based on the data as shown by Talmon (2013), the reduction of the solids effect of 5565% with method 1 with damping is overestimating the reduction, while the two methods without damping seem to underestimate the reduction, assuming that the reduction measured is caused by the effect of a lubricating viscous sub-layer. If damping is added to the Nikuradse (1933) mixing length equation, method 3, the same results are obtained as the Prtl mixing length equation with von Driest (Schlichting, 1968) damping, method 1. Apparently the mixing length damping dominates the difference between the 3 methods. The von Driest modification is an empirical damping function that fits experimental data, also changes the near-wall asymptotic behavior of the eddy viscosity t, from z2 to z4. Although neither of them are correct (DNS-data gives t proportional to z3), the von Driest damping generally improves the predictions. It has, since its first appearance, repeatedly been used in turbulence models to introduce viscous effects in the near-wall region. The von Driest damping however has never been developed to deal with the problem of a lubricating viscous sub-layer as is elaborated in this chapter. Because of the overestimation of methods 1 3 the underestimation of methods 2 4, an average between Prtl without damping, method 3, Prtl with damping, method 1, could be used according to (with h about 6.7): Since the above solutions are not (very) sensitive for changes in the pipe diameter Dp or the line speed vls, but mainly for changes of the density ratio m/l, a comparison is made for a Dp = 1 m diameter pipe at a line speed of vls = 5 m/s in s with a solids density of 2.65 ton/m3 a virtual solid with a density of 10 ton/m3. Fig. 9 Fig. 10 show the Darcy-Weisbach m 1 = l 1 ln (1 + Rsd Cv ) + h Rsd Cv l + 1 8 (30) m/l & E vs. Volumetric concentration Cv m/l: Law of the Wall (no damping) m/l: Nikuradse (no damping) m/l: Prtl (damping) m/l: Average m/l & E (-) E: Law of the Wall (no damping) E: Nikuradse (no damping) E: Prtl (damping) E: Average © S.A.M. Fig. 9. The Darcy-Weisbach friction coefficient ratio m/l the factor E as a function of the volumetric concentration Cv at Dp = 1 m, vls = 5 m/s solids density of 2.65 ton/m3. m/l & E vs. Volumetric concentration Cv m/l: Law of the Wall (no damping) m/l: Nikuradse (no damping) m/l: Prtl (damping) m/l: Average m/l & E (-) E: Law of the Wall (no damping) E: Nikuradse (no damping) E: Prtl (damping) E: Average © S.A.M. Fig. 10. The Darcy-Weisbach friction coefficient ratio m/l the factor E as a function of the volumetric concentration Cv at Dp = 1 m, vls = 5 m/s solids density of 10 ton/m3. The results of this equation are also shown in Fig. 9 Fig. 10. For ss with a solids density of 2.65 ton/m3 this gives a reduction of about 3545%, on average 40%. The downside of this equation is, that the equation gives a fixed result for a fixed Rsd·Cv value is not adaptable to more experimental data. Reason to investigate the possibility of applying a concentration profile, where the concentration equals zero at the pipe wall increases, with a sort of von Driest damping function, to a maximum value at the center of the pipe. This concentration profile has to be corrected, based on numerical integration, to ensure that the average concentration matches a given value. METHOD 5: APPLYING A CONCENTRATION PROFILE TO METHOD 2 The original Talmon (2013) concept assumes a constant density ratio for the whole cross section of the pipe. This of course is not in agreement with the physical reality. The concept assumes carrier liquid in the viscous sub-layer mixture in the remaining part of the cross-section, but uses a constant density ratio. In order to correct this a damping factor for the density ratio is proposed. This density ratio damping factor takes care that there is only carrier liquid very close to the wall. The factor A determines the thickness of this carrier liquid layer. If A equals zero, the solution obtained with Prtl or Nikuradse with von Driest damping is found, methods 1 3. If A equals 4.13 the solution of the "Law of the Wall" is found, method 4, if A equals 3.02 the solution of the Nikuradse equation without damping is found, method 2. The concentration profile the density ratio are defined as: z -A 1 - e Cv, z = Cv,max 1 + Rsd Cv, z = 1 + Rsd Cv m 1 = 2 l AC m l v ln 8 + 1 l (36) where ACv depends on the value of A. The relative excess hydraulic gradient Erhg is now: Erhg = (31) im - il = E il Rsd Cv The maximum concentration Cv,max in the concentration profile is found by integrating the concentration profile over the cross-section of the pipe making it equal to the average concentration multiplied with the cross section of the pipe according to: AC 1 + Rsd Cv - v ln m l + 1 l 8 = il 2 AC Rsd Cv v ln m l + 1 l 8 (37) Cv, z 2 ( R - z ) ( R - z ) (32) z R - A = 2 Cv,max 1 - e 0 Now, from numerical solutions, it appears that equations (36) (37) give a very good approximation of all 4 methods for the range of parameters as normally used in dredging. The factor ACv=1 for the "Law of the Wall" (method 4), ACv = 1.25 for the Nikuradse solution without damping (method 2) ACv = 3.4 for the Prtl Nikuradse solutions with von Driest damping (methods 1 3). The average equation (30) has a coefficient of ACv = 2.2 A = 1.05. Table 1 gives an overview of these values. Table 1. Some A ACv values. A Law of the Wall Nikuradse (no damping) Prtl (damping) 4.13 3.02 0.01 1.05 5.43 1.67 ACv 1.00 1.25 3.40 2.20 0.80 1.80 = Cv R2 The maximum concentration Cv,max is now equal to average concentration Cv times a correction factor. Cv,max = Cv R2 R 2 - A R R 2 - + 1- e 2 A A (33) Average Lower limit of data Upper limit of data The velocity gradient, including the concentration profile, is now: u = z - l + ( l ) + 4 2 ( ) (34) 2 2 The integrated velocity difference um-ul is now: Fig. 11 shows a lower limit of the data, an upper limit of the data the curve of Talmon (2013), method 1, compared with experimental data of Talmon (2011) in a vertical pipe. The lower upper limit are determined for the particles from d = 0.345 mm to d = 0.750 mm. The finest particles of d = 0.125 mm show less or even a reversed influence, probably because of the Thomas (1965) viscosity effect. Fig. 1 shows experimental data of Thomas (1976) of iron ore in a horizontal pipe, where the theoretical curve contains both the Thomas (1965) viscosity equation (37) with ACv = 1.3, the average of the lower upper limit. CONCLUSIONS vls, m - vls,l um - ul - l + ( l ) + 4 2 ( ) (35) - l + ( l )2 + 4 2 ( ) For the resulting Darcy-Weisbach friction factor ratio this can be approximated by: The concept of Talmon (2013) is applicable for determining the pressure losses in the homogeneous regime, however this concept has to be modified with respect to the shear stress distribution, the concentration distribution a check on conservation of volume flow concentration. The resulting equations (36) (37) with ACv = 1.3 give a good average behavior based on the data of Talmon (2011) Thomas (1976). The original factor ACv = 3.4 of Talmon (2013) seems to overestimate the reduction of the solids effect. It should be mentioned that the experiments as reported by Talmon (2011) were carried out in a vertical pipe ensuring symmetrical flow. m/l vs. Volumetric concentration Cv Theory ACv=0.8 Theory ACv=1.8 Talmon ACv=3.4 (Prtl with damping) d=0.125 mm m/l (-) d=0.345 mm d=0.560 mm d=0.750 mm © S.A.M. Fig. 11. Talmon (2011) data compared with the theory. Cv,z/Cv,max vs. Distance to the wall z/ Law of the Wall A=4.13, Acv=1.00 Nikuradse (no damping) A=3.02, Acv=1.25 Prtl (damping) A=0.01, Acv=3.40 Average A=1.05, Acv=2.20 Lower limit of data A=5.43, Acv=0.80 Upper limit of data A=1.67, Acv=1.80 Cv,z/Cv,max (-) Distance from the wall z/ (-) © S.A.M. Fig. 12. The concentration distribution for the cases considered from Table 1. Dp = 1 m, vls = 5 m/s, = 0.088 mm. For horizontal pipes the results may differ, since the velocity concentration profiles are not symmetrical at the line speeds common in dredging. The concentration profiles are shown in Figure 12. Since the model is based on a particle free viscous sub-layer the viscosity of the carrier liquid, it may not give good predictions for very small or large particles. Very small particles may influence the viscosity, while very large particles are not influenced by the viscous sub-layer. Using the resulting equations (36) (37), implies using von Driest damping in combination with a concentration profile. The resulting equations (36) (37) are flexible in use. The error of using ACv = 1.3 is difficult to define. With respect to the relative excess hydraulic gradient Erhg the accuracy is about ± 10%. With respect to the hydraulic gradient im, which is of interest for the dredging companies, the accuracy is better to much better, since this hydraulic gradient equals im = il + Erhg·Rsd·Cv. Equation (37) is implemented in the Delft Head Loss & Limit Deposit Velocity (DHLLDV) model with a default value of ACv = 1.3 (see Miedema Ramsdell (2014)).
Journal of Hydrology and Hydromechanics – de Gruyter
Published: Mar 1, 2015
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