Comparison of Maximum Allowable Pump Speed in a Horizontal and Vertical Pipe of Equal Geometry at Constant Power

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1 Pak. J. Engg. & Al. Sci. Vol. 16, Jan., 015 ( ) Comarison of Maximum Allowable Pum Seed in a Horizontal and Vertical Pie of Equal Geometry at Constant Power D. Bashar 1, A. Usman, M. K. Aliyu 3 and D. S. Yawas 3, 1. Shell Chair Office, Deartment of Mechanical Engineering, Ahmadu Bello University, Zaria.. Deartment of Chemical Engineering, Ahmadu Bello University, Zaria 3. Deartment of Mechanical Engineering, Ahmadu Bello University, Zaria. Corresonding Author: bdanasabe@abu.edu.ng Abstract This work determines and comares the maximum ossible velocity uids can be transorted through a vertical and horizontal ie of the same geometry at constant um ower. The analysis involves simulation of theoretical iing equations, analysis of the uming, frictional and static ressure with resect to ow velocity using MS excel and Matlab. Comutational results obtained from MS excel and reresented grahically in Matlab establishes the maximum allowable velocity below which the uid can be transorted safely within the required ow inut conditions (i.e. diameter, length and um ower) and above which it will be imossible for the uid to be umed to the given height or length under the same ow conditions. It was also found that the maximum allowable velocity is always greater in a horizontal ie than in a vertical ie but with uming ressure higher in the later. In addition, a general equation was determined for aroximate determination of the maximum uid velocity given ie length (height), um ower and ie diameter. Key Words: Maximum allowable velocity; Vertical height; Frictional ressure; Puming ressure. 1. Introduction Pum and iing design is very imortant economically (to the oil and gas industries) and olitically (to the oil roducing nations)as the networks of ies in the US, Euroe and Russia are resectively about 500,000km, 400,000km and 300,000km[1].Pielines are categorized based on their usage that includes; gathering ies- ies with ies large diameters at high ressure of about 4.93MPa, transortation ielines-for transmitting roduct from source to suly with ressure of between MPa and distribution ielinesconveys roducts to customers at ressure not greater than 1.4MPa [].Centrifugal ums are used for low (to moderate) ressure and high ow (high velocity) alications. The ow characteristics of a centrifugal um changes as the ressure changes. Increasing the number of vanes of an imeller in a centrifugal um will increase ow and reduce ressure and decreasing the number of vanes of the imeller will reduce ow and increase ressure [3]. Most industrial ums are oerated at high horse ower. Positive dislacement ums [4] classified as rotary um tyically work at ressures u to 5 bars (.5MPa) and recirocating ums with ressures u to 500 bars (50MPa). Centrifugal ums are referred for moving large volumes of liquid at moderate ressure while ositive dislacement ums are desired for moving small volume of uids at higher line ressures [5]. Positive dislacement ums are used for roducts with high viscosity such as heavy crude oil, bunker fuels and ashalt while centrifugal ums are used for light viscosity liquids such as water, gasoline, diesel oil and very light crude oil. A tyical 6-stage centrifugal um running at 3500 rm can roduce a differential ressure of u to 395si (.7MPa) along the ie length [6]. Also, an industrial centrifugal um with a caacity of 30 H (171.5KW) running at 300 rm can discharges water at a rate u to 340 m 3 /hr [7]. Large centrifugal ums of over 100 H(73.6 KW) can be comared with ositive dislacement ums for the same service [8]. The aim of this simulation is to comare maximum ow velocity in horizontal and vertical ies of equal geometry and to otimize um 110

2 Comarison of Maximum Allowable Pum Seed in a Horizontal and Vertical Pie of Equal Geometry at ower(for centrifugal ums) so that uid ow in ies can be transorted within the otimum ermissible ow velocity and at a comaratively lower um ower. This will enhance energy ower savings where unnecessary high horse ower ums may not be needed since comaratively lower horse ower ums can still rovide the same functions the former can rovide.. Methodology.1 Theoretical Background Flow at any oint within a ie is reresented by the general Bernoulli s energy equation [9] as: Where P V v m c g 1 PV mv hg mc constant (1) = Pressure = Volume of Flow = Flow velocity = Mass of uid = Secific heat caacity of uid = Acceleration due to gravity, and = Temerature Equation 1 imlies that the sum of the uid energy, kinetic, otential and internal energy is constant at any section of the ie. This also satisfies the conservation of energy rincile law. When the temerature is constant, internal energy becomes zero and equation (1) becomes: Pum 1 PV mv hg constant () Pie 1 h= Pie length Pum Fig.1: Pums transfering uid to a vertical and horizontal ath 1 Pie Pie length Fig. 1 Pums transferring uid to a vertical and horizontal ath Bernolli s equation for real uid can be written as [8]; 1 P 1 1 v1 z1 v g z h g g P g L (3) But since the ie diameter is constant from 1- for the vertical ie, v1 v and equation (3) becomes: P 1 P z g 1 z g P 1 P (z g g z1) 1 P g(z z1) gh L h h L L P (4) V But L o D g h K and ie length i ( z z1) hl (ie length = elevation height), we have: V Ko gh P P (5) 1 In uming uid to a certain height (Fig.1), the total ressures needed to be overcome are the frictional ressure dro and ressure due to static height (equation 5). These are designated reaction ressure in this research work. The uming ressure must be equal to or greater than this reaction ressure for the uid to move to the required height and above. These can be reresented in equation as; P (P1 P ) or P Pf, imlying the (total) uming ressure: P V Ko gh (6) For the horizontal ie in Fig. 1, elevation, ( z z1) h 0, we have: V Ko P (7). Obtaining Relevant Inut Data Relevant inut data values obtained where tabulated aroriately in Table 1. Densities and kinematic viscosities of uids were obtained aroriately [11]. Water is used as uid medium in the analysis. Other uids can also be used from the general equation determined in equation 11. Initial calculation of Reynolds number indicates that the ow is turbulent for all the velocities considered. The kinematic uid viscosity of water is found to be 1.10e7m /sat 0 C [1]. 111

3 Pak. J. Engg. & Al. Sci. Vol.16, Jan., 015 Table 1 Inut Parameters for determination of maximum velocity Sr. Data Symbol Inut Comment No. Value 1.0 Flow (average) v.0 Varied velocity (m/s) accordingly.0 Pie length (m) l 50 Varied (100, 400) 3.0 D 100 Varied i Pie diameter (mm) (40, 10) 4.0 Pum Power (KW) P 5 Constant 5.0 Pum Efficiency ç 90 Constant 6.0 Fluid Kinematic 1.10e-07 Constant viscosity of water at 0 C (m /s) 7.0 Fluid Density of water at 0 C (kg/m 3 ) Constant 8.0 Minor loss K 1 Constant o coefficient-ie i.e. iing fittings 9.0 Secific roughness factor (mm) For steel 3. Relevant Assumtions The design is considered only for the internal ow conditions only i.e. the ideal and minimal conditions required to safely um uid to a certain height without abnormal and occasional inuences, hence ie thickness is not considered. Fluid temerature is assumed constant. Therefore internal energy and vaour ressure are Table : Comutation of result outut 11 neglected. Though if resent they do not imede um ressure but ultimately add to it which relieves um ower. Their inclusion is relevant when in actual ie design i.e., ressure integrity analysis [13] where thickness has to be comensated for the inclusion so that the ies do not exlode in oeration. Effects of ressure transients due to sudden and gradual conditions are also neglected since they fall outside the normal or ideal requirement of lifting uid to certain height. This haen as a result of sudden or gradual closure of valves attached to iing systems where uid ressure subsequently increase u to the seed of sound and could have devastating consequences on iing system [14]. Occasional loads as a result of snow, earthquake, external loadings and other harsh environmental conditions are also neglected. Water is assumed to be the uid medium. Flow medium is also assumed incomressible to simlify theoretical analysis [15]. Atmoshere ressure is neglected. Mild steel is assumed to be the iing material. 4. Simulation of Theoretical Equations Table shows the comutation of ow arameters, frictional factor, uming ressure and total reaction ressure to be overcome. Chen friction factor (more exact) aroximation equation was used Sr.N Formula Units Results 1 D V m 3 /s i Flow rate: V ; 4 D V 1; e e-04 i Reynolds number and relative ie roughness factor: R ; 3 Friction factor laminar or turbulent ow: 5 6 f.0log K Re log V Head Loss: L o D g h K Pum ressure: P ; i P v K f ; R e Re 7 Pressure due to ie frictional loss and atmosheric (static) ressure: Pf hlg;ps lg 8 Total reaction ressure for vertical and horizontal ie: P P P ; P P fs f s fh f e 4 Turbulent m.0 KPa KPa; KPa KPa

4 Comarison of Maximum Allowable Pum Seed in a Horizontal and Vertical Pie of Equal Geometry at in calculation of friction factor for turbulent ow [16]. The head loss formula in Table (5) is alicable for both laminar and turbulent ows [14].Table deicts a single simulation result (a oint coordinates in Fig. ) for the inut arameters of Table Result Analysis In rincile, the total uming ressure must be greater than the total reaction ressure. The grah of the uming ressure (reresenting thick line),reaction ressure for vertical ie (reresenting dashed-dotted lines), reaction ressure for horizontal ie (reresenting dashed lines) and static ressure (reresenting dotted line) are therefore lotted against varying ow velocity. From Fig., it can be seen that the uming ressure decreases and the reaction ressures (in both vertical and horizontal ie) increases as the velocity increases until a oint is reached when they are equal and above which the reaction ressure dramatically exceeds the uming ressure. The velocity at which the uming and reaction ressure are equal is the maximum velocity the uid can attain at secified height given the required ow conditions. The allowable ow velocity range is when net ressure (difference between uming and reaction ressure) is ositive. For Fig., the maximum ossible velocities for the vertical and horizontal ath are at 1.15m/s and 4.8m/s and the allowable velocity ranges are between 0 < v <1.15 and 0 < v <4.8. The frictional ressure loss (difference between the dotted and the dashed lines) is significant only within the allowable velocity range for the horizontal ie. Similarly, from Fig. 3, the maximum ossible velocities for both ie aths are at 3.59m/sand 5.87m/s and the allowable velocity ranges are between0 < v < 3.59 and 0 < v < The frictional ressure losses are both significant (difference between the dotted and the dashed lines) within the allowable velocity range. Also from Fig. 4, the maximum ossible velocities for both ie ath are at 0.47m/s and.7m/s and the allowable velocity ranges are between0 < v < 0.47 and 0 < v <.7 resectively. The frictional ressure loss here is only significant within the allowable velocity range for the horizontal ie. Frictional ressure loss for the vertical ie of Fig. and Fig. 3 can be neglected since the dasheddotted and the dotted lines meet the thick line at aroximately the same velocity (ignoring frictional ressure means the only reaction ressure to be overcome is the static ressure which is constant). However, the frictional loss cannot be ignored for Fig. 3 and horizontal ie ath of Fig. and 4 since the dashed and dotted lines meet the thick line at different maximum velocities. In other words, the maximum allowable ow velocity in Fig. 3 can only be 3.59m/s and not 4.75m/s (intersection of the dashed-dotted line with the thick line) for the vertical ie ath. However, the effect of change in ie length and diameter resectively is also considered. It can be noted that as the diameter (from Fig. at 100mm) decreases to 50mm in Fig. 3, the uming ressure changes (from 1 P 57.7 v to P 1 91v and the maximum allowable velocity increases and consequently the velocity range (as in shown in Fig. 3). It can be seen that as the length (from Fig. at 50m) increases to 500m in Fig. 4, the maximum allowable velocity decreases and consequently the velocity range (as in shown in Fig. 4) but the uming ressure is maintained (i.e. 1 P 91v ). All the three grahs are roduced at constant ower of 5KW. This shows that ums can be otimized (redesigned or reengineered) at lower ower to transort liquid at much longer distances. 5.1 Determination of the Maximum Fluid Velocity from Simulation The maximum allowable ow velocity can also be calculated using equations generated for Figures, 3 and 4 in Matlab (curve fitting tools). The intersection of the total action and reaction ressures gives an equation in form of velocity which can be calculated as given below: 56.7v v vertical ie ath (from Fig.) 56.7v 56.7v v 4.86v 1.v v v 0.3 horizontal ie ath (from Fig.) 56.7v v 1.v0.3 0 for for the (8) the (9) 113

5 Pak. J. Engg. & Al. Sci. Vol.16, Jan., 015 Solving using Matlab gives three roots; one real number and comlex numbers. The real number is the maximum allowable velocity. The real and corresonding comlex roots for the vertical and horizontal ath resectively are; ( 1.15 v) 0, imlying v and {( 10.09i 0.70) v} 0, imlying v ( 10.09i 0.70) ( 4.8 v) 0, imlying v 4. 8 and {( 4.5i.53) v} 0, imlying v ( 4.5i.53) Similarly for Figure 3, after simlifying, the real and corresonding comlex roots for the vertical and horizontal ath resectively are; ( 3.59 v) 0, imlying v and {( 7.44i 1.93) v} 0, imlying v ( 7.44i 1.93) ( 9.87 v) 0, imlying v and {( 5.17i 3.07) v} 0, imlying v ( 5.17i 3.07) Similarly for Figure 4, after simlifying, the real and corresonding comlex roots for the vertical and horizontal ath resectively are; ( 0.47 v) 0, imlying v and, {( 6.867i 0.35) v} 0, imlying v ( 6.86i 0.35) (.7 v) 0, imlying v. 7 and {(.4i 1.48) v} 0, imlying v (.4i 1.48) 5. Determination of Maximum Velocity Using Derived Equation From equation 6, the total uming ressure must be greater than the reaction ressure (From Table, Items 8 and 9) written as; P Av P Av P P P P P (10) fs 1 Ko v lg 1 Ko v lg 0 f s simlifying; simlifying; Substituting area, 4P v 4, we have 1 Ko v lg 0 general equation can hence be written as: av, and hence a bv c0 (11) 4P v 1 o Where a ;b K ; and c lg i.e., for vertical ie and i.e. for a horizontal ie. The maximum velocity was determined using equation 11 for each of arameters in Fig., 3 and 4 and solved grahically as deicted in Fig. 5, 6 and 7. The only constraint using the equation is determining the frictional factor, f. The friction factor was calculated from an average of two reasonably assumed velocities (1 and 4m/s). Equations from simulation (Fig., 3 and 4) were lotted along side with equations generated using equation 11 as deicted in Fig. 5, 6 and 7. Equations generated from equation 11 using same arameters as in Fig., 3 and 4 for the vertical (Equation 11) and horizontal (Equation 1) ath are resectively given below: 57.7v v 57.7v 5.5v 90.9v v 5.5v 90.9v ; 10.4v ; ; 0; 90.9v 99.8v 0; v 0 (1 (13) Analysis of the maximum velocity from equation and simulation are deicted in Table 3 below: 5.3 Grahical Determination of the Real Root and Exit Pressure Using the General Equation Analysis of the maximum velocity determined using equation 11 and simulation (of Table 1 and ) results were lotted in excel to determine the real roots. asically Figs. 5, 6 and 7 are grahical method (instead of using Matlab) of finding the roots of equation 11 and simulation results (Figs., 3 and 4). 114

6 Comarison of Maximum Allowable Pum Seed in a Horizontal and Vertical Pie of Equal Geometry at Table 3: Comarison of the accuracy of the maximum velocity using equation and simulation Item Equation: (m/s) Simulation: (m/s) Percentage Error (%) Corrected Maximum Velocity for Equation: (m/s) V H V H V H V H Fig Fig Fig Note: V means vertical and H horizontal Fig. Effects of ressure with velocity for a 5kw, 50m and 100 mm um ower, ie length and ie diameter resectively. 115

7 Pak. J. Engg. & Al. Sci. Vol.16, Jan., 015 Fig. 3 Effects of ressure with velocity for a 5kw, 50m and 50 mm um ower, ie length and ie diameter resectively. Fig. 4 Effects of ressure with velocity for a 5kw, 500m and 50 mm um ower, ie length and ie diameter resectively. 116

8 Comarison of Maximum Allowable Pum Seed in a Horizontal and Vertical Pie of Equal Geometry at The exit ressure is the net ressure between the um ressure and reaction ressure. This is evaluated from Table as ( P Pfs) and ( P Pfh) for the vertical and horizontal ies resectively. Fig.5 deicts the variation of net exit ressure with increasing velocity for the simulation of Fig. i.e. at um ower of 5KW, ie length of 50mm and ie diameter 100mm. As the net ressure decreases, it reaches a oint when the um ressure and the reaction ressure are equal (net ressure is zero) and the velocity at this oint (1.15m/s for vertical ie) on the grah reresents the maximum allowable velocity as further decrease in the net ressure will result in an insignificant (negative) value. The grahical reresentation using equation 11 and simulation coincides (are equal) and with little deviation at lower negative net ressure. Similarly, Fig. 6 deicts the variation of net exit ressure with increasing velocity for the simulation of Fig. 3 i.e. at um ower of 5KW, ie length of 50mm and ie diameter 50mm. As the net exit ressure decreases, it reaches a oint when the um ressure and the reaction ressure are equal (net exit ressure is zero) and the velocity at this oint (3.59m/s for vertical ie) on the grah reresents the maximum allowable velocity as further decrease in the exit ressure will result in an insignificant (negative) value. Fig. 7 also deicts the variation of net exit ressure with increasing velocity for the simulation of Fig. 3 i.e. at um ower of 5KW, ie length of 500mm and ie diameter 50mm. As the net exit ressure decreases, it reaches a oint when the um ressure and the reaction ressure are equal (net exit ressure is zero) and the velocity at this oint (0.47m/s for vertical ie) on the grah reresents the maximum allowable velocity as further decrease in the exit ressure will result in an insignificant (negative) value. The allowable velocity range from Fig. 5, 6 and 7 for the vertical and horizontal ies resectively are given as: Fig. 5; (0 < v < 1.15);(0 < v < 4.8) Fig. 6; (0 < v < 3.59);(0 < v < 5.87) and Fig. 7; (0 < v < 0.47);(0 < v <.7) Fig. 5 Effects of Net Pie Exit Pressure with Increasing Velocity for Fig.. 117

9 Pak. J. Engg. & Al. Sci. Vol.16, Jan., 015 Fig. 6 Effects of Pie Exit Pressure with Increasing Velocity for Fig. 3. Fig. 7 Effects of Pie Exit Pressure with Increasing Velocity for Fig

10 Comarison of Maximum Allowable Pum Seed in a Horizontal and Vertical Pie of Equal Geometry at The allowable velocity range that corresonds to the aroriate net exit ressure required can be selected and it is advisable to aly a factor of safety of two-third (/3) to the calculated maximum velocity (for that using equation 11) since this will always fall within the actual velocity range (Table 3, columns 8 and 9). The selected velocity (within the allowable range) can then be used to calculate the um ressure and consequently the ow rate of the desired um. Results of Fig., 3 and 4 shows that ums (i.e. centrifugal ums) can be designed with high ressure and lower ow for vertical uid ath and low ressure and higher ow for a horizontal ow ath. 6. Conclusions Large centrifugal ums of over 100 H (73.6 KW) are used to move uids within long distances. Centrifugal ums can be designed with high ressure and lower ow for vertical uid ath and low ressure and higher ow for a horizontal ow ath. This can be achieved by either increasing the number of vanes of an imeller for increased ow and reduced ressure or decreasing the number of vanes of the imeller for decreased ow and increased ressure [3]. Analysis of um ressure and arameters in this research work shows that ums can be redesigned or re-engineered at lower ower to transort liquid at much longer distances. This will enhance energy ower savings where unnecessary large industrial ums may be substituted with comaratively lower horse ower ums and still deliver uid roducts effectively. The result of the findings shows that: At any constant um ower and ow requirements (i.e. diameter, length and uid medium), a maximum allowable velocity is established below which the uid can be transorted safely to the required height or length and above which it will be imossible for the uid to be umed to the given height or length under the same ow conditions The maximum ermissible velocity is always greater in a horizontal ie than in a vertical ie laced erendicular to ground but a higher uming ressure is always required in the later. A general equation is established that determines the maximum allowable ow velocity for ums given the required length, diameter and um ower. The equation has three roots- a real and corresonding two equal comlex roots. The real root corresonds to the maximum allowable ow velocity: a v 4P bv c0 1 o Where a ; b K ; clg vertical ie and c = 0 i.e., for a horizontal ie. i.e., The solution of the general equation is an aroximate one with a ercentage error of not greater than ±4%. This is because the friction factor cannot be accurately calculated at each varying velocity. The friction factor was calculated from an average of two reasonably assumed velocities (1 and 4m/s). It will also be advisable to aly a factor of safety of /3 to the calculated maximum velocity since this will always fall within the actual velocity range; 0 v d Where d is the real root from the 3 general equation. It is advisable to lot the general equation in either MS excel or Matlab in order to choose the aroriate velocity (within the allowable velocity range) that corresonds to the required net exit ie ressure so desired. At constant um ower, as the diameter decreases, the maximum allowable velocity increases and as the length increases the maximum allowable velocity decreases. Therefore the greater the ie length and diameter, the lower the maximum allowable ow velocity for each resectively. Conversely, the lower the ie length and diameter, the greater the maximum allowable ow velocity and its range resectively. Inclusion of frictional ressure dro in iing system is very significant as neglecting it sometimes may result in choosing the inaroriate ow velocity. 119

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