1. Introduction. Volume 7 Issue 1, January Licensed Under Creative Commons Attribution CC BY

Transcription

1 turbulent flow in periodically small pipes. Here we discussed a universal resistance equation relating friction factor number (Re) and roughness height () for the entire range of turbulent flow in small pipes covering regimes in smoo have also discussed about the variation in friction factor F, used in the arcy Formula with the Reynolds number in turbulent flow. As discussed Colebrook-White formula deviates from Nikuradse experimental results in transitio difference in roughness factor. Hazen-Williams equation and review of equations on friction factor to be used. Su found to be sufficient to predict the friction factor for all ranges of Re and different values of (). Keywords: Friction factor, Turbulent flow, Reynolds number, arcy formula. Colebrook equation, Hazen-Williams equ. Introduction To study the differentiation in friction factor, F, used in the arcy Formula with the Reynolds number in both laminar and turbulent flow. The friction factor will be temperance as a function of Reynolds number and the roughness will be calculated using the Colebrook equation. The loss of head resulting from the flow of a fluid through a pipeline is demonstrated by the arcy Formula. In this equation (), whereh f the loss of head (units of length) is and the average velocity is V. The friction factor, F, varies with Reynolds number and a roughness factor, L is the characteristics length of the pipe, is the diameter of the pipe, g is the acceleration due to the gravity. The friction factor (F) is a measure of the shear stress (or shear force per unit area) that the turbulent flow exerts on the wall of a pipe. The Hagen-poiseuille equation for laminar flow declares that the head loss is the unrestricted of surface roughness. Thus in laminar flow the head loss varies as average velocity v and inversely proportional to the pipe diameter 2.Now comparing equation () and (2) we can get, We know that, R= ρv μ From (3), declaring that the friction factor is amount to viscosity and conversely amount to the velocity,pipe diameter and fluid density under laminar flow conditions.the friction factor is unrestricted of pipe roughness in laminar flow because the disorders motive by surface roughness are hurriedly humored by viscosity. Equation (2) can be solved for the pressure drop as a function of total discharge to receive. When the flow is turbulent the connection complex and is best shown by means of a friction factor is a function of both Reyno roughness. Where as in turbulent flow(r>>4 factor, depends upon the Reynolds numbe relative roughness of the pipe, Where, roughness height of the pipe. The usual treat pipe flow in appearance of surface roughnes Nikuradse showed the dependence on rou pipes not naturally roughened by bindin uniform sand-grains to the pipe walls. roughness was specified as the ratio of diameter to the pipe diameter ( ). When likened to the pipe diameter.i.e. 0, F Reynolds number R. The connection betw factor and Reynolds number can be firm m comparative roughness. When is of a sig low Reynolds number R, the flow can be smooth regime (there is no effect of connection between the friction factor and R can be resolute for every comparative r increase, the flow becomes transitionally transition regime. From these connections, i for rough pipes the roughness is more im Reynolds number in resolutions the magnitu factor. At high Reynolds number (complete t pipes).the friction factor depends wholly o the friction factor can be received from the ro In a smooth pipe flow, the viscous sub laye the effect of on the flow. In which the fric above the smooth value and is a function o flow ultimately arrives a wholly rough regim unrestricted on R. In this case, the frictio function of R and is unrestricted of the eff flow. Hence the smooth pipe law is. Volume 7 Issue, January 209 Paper I: IJSER8522

2 ascertained practicality of Colebrook-white equation over a wide range of Reynolds numbers and relative roughness value, the equation unanimously adopted is due to Colebrook and white (937)[9 proposed the following equation. Equation (9) veils not only the transition region but also the wholly developed smooth and rough pipes. By putting 0, Equation (9) reduces to equation (7) for smooth pipes and as R ; equation (9) becomes equation (8) for rough pipes. Equation (9) has become the act of receiving unit of measurement for calculating the friction factors. It suffers; however, from being an implicit equation in F and thus requires an iterative solution. Their calculation results were however, thoroughly different from those received in the laboratory when using the Colebrook-white equations. The friction factor determined from laboratory data decrease with an increase in the Reynolds number even after a perfectly sure critical value, where as the friction factor of the Colebrook-white equation tends to be constant with an increase in the Reynolds number. The Colebrook equation can be used to resolute the absolute roughness,, experimentally measuring the friction factor and Reynolds number. Since the mid-970s, many alternative distinct equations have been developed to abandon the iterative procedure instinctive to the Colebrook-white equation. Alternatively the distinct equation for the friction factor derived by swam me and gain can be solved for the absolute roughness. When solving for the roughness it is important to note that the quantity in equation () that is squared is negative. Equation (0) and (2) are not equivalent and will yield slightly different results with the error a function of the Reynolds number. ( γ u and) with a rectification factor, reportin i.e. smooth, transition, and rough regimes of thus Where,R is the friction Reynolds number a to u.atr γ 0, pipe is said to be in smoo R pipe is said it be in rough condition For large values of γ, the term γ domin u u second term b negligible in resemblance w small values of γ, the second term bec u allowing the neglect of the first term. Thus reporting all the regions can be compressed Now, if a condition that φ(r )= for both w R is imposed, equation (6) reduces and equation (4) respectively. From th Equation (6) can be exchanged i 8( u u )2 for the friction factor reporting the whole ra flows. Thus the hindrance equation for pip smooth, transition and rough regimes can be By dissolved Nikuradse s data on measurements in gravel roughened pipes values of a= (0.333) and b= (0.0) has φ(r ) is given by The legality of the manifestation for B alon shown in Figure () by using the Nikurads data 2. Proposed Model The established laws of velocity distribution for turbulent flows are given by, Paper I: IJSER8522 Volume 7 Issue, January 209

3 data on the gravel roughened pipe is shown in uring the past year since Moody s chart the equations on friction factor have appeared as. Moody (947): He proposed the equation = (+ ( Re ) /3 ) 2. Wood (966): It is valid for Re>0000 an Figure 2: Friction factor diagram The hindrance equation, as given by equation (7) pleasurably fits the entire data of Nikuradse s on sand roughened pipes for varying relative roughness heights. In addition to Nikuradse s experimental data, resistance equation is also conspired for the most current experimental pipe friction data on smooth pipes (McKeon.et al, 2004).Thus a universal resistance equation is developed in the form of equation (7). =.094( ) ( ) +88( ).44.R -. Where φ =.62( ) o Eck (973): He proposed the equation as: = -2log ( ) Re 4. Churchill (973): He proposed the equatio = -2log (( ) + ( 7 ) 0.9 ) 3.7 Re 5. Jain and Swamis (976): They propos covering the range of Re from to 0 between and 0.05 as: of =-2log ( Re 0.9) 6. Jain (976): He proposed the equation as: = -2log (( ) (6.943) 0.9 ) Re Figure 3: Friction Factor diagram Moody Chart Figure 4: Effect of wall roughness on turbulent pipe flow 7. Churchill (977): The author claimed t holds for all Re and has the following: = 8[( 8 ) 2 + Re (A+B).5/2 Where, A= [ ln [( 7 B= ( Re ) 6 Re ) Chen (979): He also proposed equation f covering all the ranges of Re and. = -2 log [ Re log ( ( ) 9. Round (980): He also proposed the following form: Volume 7 Issue, January 209 Paper I: IJSER8522

4 2. Haalland (983): He proposed a variation in the effect of the relative roughness by the following expression: = -.8 log [( ) Re 3. Serghide s (984): He proposed the equation in the following expression: ) 2 = [ ( Or = [4.78 ( 4.78) Manadilli (997): He proposed the following expressions valid for Re ranging from 5235 to 08 and for any value of. = 2 log [ Re Re 5. Monzon, Romeo, Royo (2002): They proposed the equation in the following expression: =-2log[ log ( Re log(( Re ) ( Re ) )) 6. Goudar, Sonnad (2006): They proposed the equation in the following expression: = ln [.4587 Re s (S 0.3) s+ Where, S=0.24Re +ln (0.4587Re) 7. Vatankhan, Kouchakzadeh (2008): They proposed the equation in the following expression: Re = ln[ s (S 0.3) s Where, S=0.24Re + ln (0.4587Re) 8. Buzzeli (2008): He proposed the equation as: =α [α+2log β ( Re ) β ln Re.4 Where, α= [ β= 3.7 Re+2.5α +.32 substitution method to the Colebrook-whi Colebrook curve corresponding to s=7.5 μm Colebrook curve makes a needy pred transitionally rough treatment of this surface going out from the smooth regime, At Red Colebrook relation over evaluates the fr approximately 0%. In the transitional re following the Colebrook correlation, The term national roughness, similar to the t gravel corn roughness flavored by Nikurad of the fact that neutered surfaces are often Colebrook s terms, natural or commer More purity can be achieved by using a internal substitutions to the Colebrook-whi a new distinct formula for calculating the fr a given relative roughness, the Nikurad correlation. The Colebrook curves for the e corn roughness. In the transitional regim between the Colebrook curves and those ca method used here are significant. Whereas roughness displays a term national connection, the Colebrook curves monotonic the smooth curve and draw near the who from above. As discussed, Colebrook deviates from Nikuradse experimental resu range, because of their difference in r Colebrook-white formula is for irregular su in pipes resulting from the growing procedu flow, the friction factor correlations are m they are implicit in F. For turbulent flow the unanimous law of friction factor relate turbulent flow in rough pipes which is of interest, the Colebrook-white equation is widely used correlation to calculate F. It re factor to the Reynolds number and pipe present a book of fiction,mathematically e of the Colebrook-white equation to calcula for turbulent flow in rough pipes. This new no iterative calculations are precise discrimination. A limiting case of this eq friction factor computes with a maximum and a maximum percentage error grid of and Re values ( 0 2 ; R 0 8.This was more best recently available non-iterative appro Colebrook-white equation(maximum abs 0.058;maximum percentage error of.42%) equally valid for commercial pipes and pipes. By making correction factor φ(re ) Volume 7 Issue, January 209 Paper I: IJSER8522

5 Figure 6: Prediction for commercial pipe Figure 7: Percentage of error in the estimation of Colebrook-white formula. with Example: (use of moody iagram to find friction factor): A commercial steel pipe,.5 m in diameter, carries a 3.5 m 3 /s of water at 20 0 c. etermine the friction factor and the flow regime. (I.e. laminar-critical, turbulent-transitional zone, turbulent-smooth pipe or turbulent rough pipe).sol: To determine the friction factor, relative roughness and the Reynolds number should be calculated. For commercial steel pipe, roughness height () = mm Relative roughness ( ) =0.045 mm/500 mm= V=Q/A=(3.5 m 3 /s)/[ π (.5 4 m)2 =.98 m/s N R = v /v=[(.5 m)(.98 m/s) / ( m 2 /s) = From moody iagram, f=0.0 The flow is turbulent-transitional zone. As discussed, proposed model predicts reasonably well in the entire turbulent ranges of pipe flow and equally valid in case of commercial pipes as well as sand roughened pipes, this can be used as an alternative of Hazen-Williams formula in designing the pipe line. pipes made of different materials. In most state of rest engineers use the Hazen-Will characterize the roughness of the pipes Relatively precise results can be received personal pipes of a looped procedure by skilled supposition for the flow in each pipe plinth on continuity and then using succes with new flow (Q) values received by the equation. Commercial software has been de solve very intricate procedure using the draw near as well as other technically deve It was developed for water flow in large approximately 2 in) within an abstemious velocity (V3 m/s, approximately 0 ft/s). technical, the Hazen-Williams equation is n homogeneous and its ranges of practicality 998).Hazen-Williams equation, originally British durations procedure, has been form, V = K C HW R 0.63 S E 0.54 Where, V=Mean fluid velocity in the pipe, units and K=o.85 for S.I units. C HW = Hazen-Williams resistance coefficien state of rest radius (A/P, Where A is the cro and P is the wetted perimeter) S E = Slope of the energy grade line or the h length of the pipe (S=h/L). By making use of equation (), equation ( Williams formula. C can be inter R 0.06 ( ) γ 0.08, is imp function of R,, and Kinematics viscos found to be dependent on pipe diameter (Lie 5. Application of the Proposed Model Connection between Hazen-Williams C and rest parameters. iscrimination of head losses due to friction in pipes Figure 8: Variations in C important task in optimization studies and when in a state of Volume 7 Issue, January 209 Paper I: IJSER8522

6 Cast-Iron(common in older water line) New unlined 30 0-year-old year-old year-old year-old Concrete or Concrete-lined Steel forms 40 Wooden forms 20 Smooth 40 Average 20 Rough 00 Centrifugally spun 35 Copper Table of Roughness Heights, for certain common materials Pipe materials (mm) (ft) Brass Concrete Steel forms, Smooth Good join is average Rough visible form marks Copper Corrugated metal(cmp) Iron(common in older water lines except ductile or IP) Asphalt lined Cast uctile, IP-cement mortar lined Galvanized Wrought Polyvinyl chloride (PVC) Polyethylene high density(hpe) Steel Enamel coaled Riveted Seamless C0mmercial [5 Von Bernuth, R.., and Wilson, T., Fr small diameter plastic pipes.j.hydraw.e 92, 989. [6 Wood,.J., An Explicit f relationship.civileng, 60-6, 966. [7 Round, G.F., An explicit approximatio factor Reynolds number relation for ro pipes. Can.J.Chem.Eng. 58,22-23, 9 [8 Reynolds.(883):Phil.Trans.Roy.Soc.,L [9 Hultmark, Marcus, Reynolds num turbulent pipe flow. Mechanical eng 20. [0 Kwok, K.C.S., Turbulence Effect o Circular Cylinder, Journal of Mechanics.2, Conclusion Plinth on the Nikuradse s experimental data, an improved version of equation on friction factor covering the whole Turbulent flow range flow has been presented. The friction factor treatment of a nectar surface in the transitional regime does not follow the Colebrook relationship and in lieu of exhibits treatment more pattern cal of Nikuradse s gravel corn roughness. Paper I: IJSER8522 Volume 7 Issue, January 209