ACCURATE, EXPLICIT PIPE SIZING FORMULA FOR TURBULENT FLOWS

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1 Journal o cience and Technology, Vol. 9, No. (009), pp Kwame Nkrumah University o cience and Technology (KNUT) TECHNICAL NOTE ACCURATE, EXPLICIT PIPE IZING FORMULA FOR TURBULENT FLOW K.O. Aiyesimoju epartment o Civil Engineering University o Lagos Lagos, Nigeria ABTRACT This paper develops an explicit ormula or computing the diameter o pipes, which is applicable to all turbulent lows. The ormula not only avoids iteration but still estimates pipe diameters over the entire range o turbulent lows with an error o less than 4% in the worst cases. This is superior to (without requiring a higher level o diiculty in use) the only two previously developed relationships that are generally applicable to turbulent lows and do not require iteration. Their errors were up to 4% or Ranga Raju s method and up to % or Prabhata and Akalank s method. All the errors are relative to the ideal but iterative Colebrook-White ormula. Keywords: Pipe sizing, turbulent low, explicit INTROUCTION The importance o accurate sizing o pipes cannot be overemphasized in pipeline engineering since an overestimation implies a high initial cost o conduit installation whilst underestimation will lead to unctional inadequacies. Laminar lows are governed by the Hagen- Poiseuille equation, which shows a linear relationship between head loss, pipe discharge and pipe diameter. This enables pipe diameters to be estimated accurately and without iteration or any given head loss and discharge. Turbulent lows however are governed by more complex relationships. Aiyesimoju (00) has compared the accuracy o the Hazen-Williams, Manning and Colebrook-White ormulas which are the most commonly used ones o the numerous semi-empirical and ully empirical riction loss relationships or turbulent lows in water supply practice and conirmed that the Colebrook-White ormula yields the best results. The problem is that computation o pipe diameters with the ormula involves iteration. Numerous graphical aids such as Moody s Chart (see treeter, 1971) and Asthana s iagram (Asthana, 1974) which were developed to simpliy using this ormula, are however oten not available and when they are, the errors introduced in scaling the graphs are oten signiicant. Aiyesimoju (008) recently developed an explicit ormula or pipe sizing but this was speciically tailored to water distribution pipes, in which case, errors are within %. Other recent work in this area has been about assessment or Journal o cience and Technology KNUT ecember 009

2 148 Aiyesimoju proper use o existing methods. For example, Christensen et al. (000) was concerned only with the proper use o the Hazen-Williams ormula whilst Khatibi et al. (000) ocussed mainly on the estimation o riction actors or low in nearly lat tidal channels. Although Aiyesimoju (006) obtained an accurate equation or estimating turbulent head losses in water distribution pipes which avoids iteration, the equation does not avoid iteration i it is to be used to estimate the pipe diameter. This paper is to develop an explicit ormula to compute required pipe diameters, which is useul over the entire range o practical turbulent low situations. The developed relationship as well as the only two previously developed relationships that are generally applicable to turbulent lows and that do not involve any iteration, will be compared with the most accurate Colebrook-White ormula. The two methods are (1) Ranga Raju and Garde equation (Ranga Raju and Garde, 1971) and () Prabhata and Akalank equation (Prabhata and Akalank, 1976). The comparison will be in the context o several examples o low situations designed to relect the ull range o turbulent lows, as ollows: 1. Pipe equivalent roughness height ε between 0.01mm (much less than the value or drawn tubing) and cm (much larger than corresponding to riveted steel).. Pipe diameter between 0.1mm and 50m.. Pipe relative roughness e/ up to 0.5 (over times the limit in Moody s Chart). 4. Flow Reynold s Number R above 4000 (lower limit or turbulent pipe low). 5. Minimum kinematic viscosity n min o luid will be taken as -8 m /s (order o magnitude less than benzene s) and maximum n max will be taken as - m /s (order o magnitude greater than glycerol s). V g (1) where the riction actor is estimated with the Colebrook-White ormula, 1.51 log ().7 R In the above equations, is the riction slope (head loss h over pipe length L), V is low velocity, is pipe diameter, g is acceleration due to gravity, ε is the pipe roughness height (a measure o the typical height o protrusions rom the pipe material surace and R is pipe Reynold s number. The Colebrook-White ormula is based on the results o experiments on sand-roughened pipes (see treeter, 1971), with appropriate modiication o the results to relect observations o commercial pipes. The well known Moody s Chart is simply a graphical presentation o this. I is pipe discharge, n is the luid kinematic viscosity and g is assumed as 9.8 m/s, then 4 V () and R V 4 Eqn. 1 implies g V 1.7 Using this in Eqn. gives ubstituting Eqns. 4 and 5 in Eqn. yields (4) (5) PIPE IAMETER ETIMATION METHO Colebrook-White Formula Friction loss in turbulent pipe low is best estimated by the arcy-weisbach ormula (treeter, 1971) log v (6) Journal o cience and Technology KNUT ecember 009

3 Accurate, explicit pipe sizing ormula Ranga Raju and Garde s method RangaRaju and Garde (1971) proposed the ollowing equations or pipe sizing a) For e g /n - b) where.6 which gives g For e g /n > -.85 which results in * is the non-dimensional diameter (g / ) 0. ε* is the non-dimensional roughness ε(g / ) 0. ν* is the non-dimensional roughness ν(1/g ) 0. g g g. 51 log R o 0.50 (7a) (7b) (7c) (7d) Prabhata and Akalank s method Prabhata and Akalank (1976) proposed the ollowing equation * 0.66( * *) (8) hand side o the equation is not very sensitive to the value o on the right hand side. This suggests that reasonably accurate riction actors can be obtained by assuming a constant value or the on the right hand side (say o ). Thus rom Equation, an explicit estimate or the riction actor is 1 log.7.51 R o (9a) The best value o o over the entire range o low o practical interest will be determined in this study. ubstituting Eqns. 4 and 5 in Eqn. 9a and rearranging, yields. 51 log.5 (9b) R o To avoid iterating or, its value on the right hand side o Eqn. 9b will be replaced by an estimate ' thus yielding.479 log.5.7'.51 R o () A good estimate or ' can be obtained assuming a riction actor o o in Eqn. 5, thus giving o (11) which implies ' can be estimated as o (1) Also, rom Eqn. 4, an approximation can be obtained or the Reynolds Number, or use on the right hand side o Eqn., as Proposed method Eqn. cannot be solved explicitly or but experience suggests that the value o on the let R 1.7 Journal o cience and Technology KNUT ecember 009

4 150 ubstituting the above in Eqn. yields log which rom Eqn. 1 can be rewritten as Aiyesimoju o log.7 0 TET PROBLEM Aiyesimoju (00) has designed eight test problems which cover the extreme range o practical low situations. These are applicable here and are thus adopted. The scenarios (where κ is ε/) are as shown in Table 1. The problem in each test case is to determine the pipe diameter or that case by the dierent methods under consideration. The perormance o a method in a test problem will be measured by the actor (relative to Colebrook-White (1a) (1b) method) within which it is able to estimate the diameter or that test problem. REULT The speciic parameters to ensure the earlier speciied range practical turbulent lows are covered by the test problems described in Table 1, are shown in Table or the ideal situation (Colebrook-White ormula). For each test problem, the diameters as well as the actors o accuracy (the ratio o the larger o the computed diameter or the method and that by Colebrook- White ormula to the smaller o the two) were computed or Prabhata and Akalank method, Ranga Raju and Garde method and the proposed method. These are all shown in Table. ICUION The actor o accuracy as deined, is unity or a perect result and the larger it is the less accurate the result is. It is o course never less than unity. The computed actors o accuracy or the proposed method in Table depend upon the value o o assumed (rom Equation 1b). etting the Table 1: cenarios o the test problems Test No k low low large large low low large large R low large low large low large low large large large large large small small small small Table : peciic low parameters or dierent Test cases Case No. Rough height (m) Pipe iameter (m) Colebrok-White (Ideal) Flow Parameters Relative Rough Reynold's Number Kinematic Viscosity ischarge Friction Factor Friction slope e k=e/ R n 1 1.E E- 4.0E+0 1.E E E-0.58E-11 1.E E- 1.0E+08 1.E-05.9E E-0.4E-0 1.E E-0 4.0E+0 1.E E+00 4.E-0.71E E E-0 1.0E+08 1.E-05.9E+04.E E E E E E+0 1.E-05.1E E-0.4E E E E E+08 1.E E-0 1.E-0 6.1E E E E E+0 1.E-05.1E-06.E-01.70E E E E E+08 1.E E-0.E E+16 Journal o cience and Technology KNUT ecember 009

5 Accurate, explicit pipe sizing ormula Table : Computed pipe diameters and actors o accuracy or dierent test cases and dierent methods Case Ranga Raju Prabhata and Akalank Proposed No. Pipe iameter (eqn.7 ) Accuracy (eqn.8 ) Accuracy (eqn.14 ) Accuracy Factor o Pipe iameter Factor o Pipe iameter Factor o E E E E E E E E E E E E E E E E E E E E E E E E value o o to 0.15 gave the least value (o 1.06) or the worst actor o accuracy or the eight test problems. The displayed results or the proposed method in Table are or this case. Although a kinematic viscosity coeicient o -5 m /s is shown in the Table, variation o this value rom -8 m /s up to - m /s had no noticeable eect on the results. A simple, accurate and explicit procedure or estimation o turbulent low pipe diameters can thus obtained by substituting o o 0.15 in Equations 1b and 1 to give the proposed method as where log (14) From Table, the worst actors o accuracy were 1.40 or Ranga Raju s method, 1. or Prabhata and Akalank s method and 1.06 or the proposed method. Thus even in these extreme cases, the proposed method estimated diameters with less than 4% error as compared with up to 4% or Ranga Raju s method and up to % or Prabhata and Akalank s method. CONCLUION An explicit ormula has been developed to compute required pipe diameters or any given riction slope and discharge, without iteration and which is useul over the entire range o practical turbulent low situations. The developed relationship (Equations 14 and 15) as well as the only two previously developed relationships that are generally applicable to turbulent lows and that do not involve any iteration (Ranga Raju s method and Prabhata and Akalank s method), were compared with the most accurate but iterative Colebrook-White ormula. Even in the extreme cases, proposed method estimated diameters with less than 4% error as compared with up to 4% or Ranga Raju s method and up to % or Prabhata and Akalank s method. REFERENCE Aiyesimoju, K.O. (00). A comparison o riction loss relationships or turbulent pipe low, Journal o Engineering Research, Vol. JER-, Nos.1&, pp. 15- Aiyesimoju, K.O. (006). impliied estimation o turbulent head losses in water distribution pipes, Technical Transactions, Nigerian ociety o Engineers, Vol. 41, No. 1, pp Journal o cience and Technology KNUT ecember 009

6 15 Aiyesimoju Aiyesimoju, K.O. (008). A non-iterative procedure or Colebrook-White estimation o water distribution pipe diameters, Technical Transactions, Nigerian ociety o Engineers, Vol. 4, No. 4, pp Asthana, K.C. (1974). Transormation o Moody iagram, Journal o Hydraulics ivision, ACE, Vol. 0, No. 6, pp Christensen, B.A., Locher, F.A. and mamee, P.K. (000). Limitations and proper use o the Hazen-Williams Equation, Journal o Hydraulic Engineering, ACE, 16(). pp Khatibi, R.H., Williams, J.J.R. and Wormleaton, P.R. (000). Friction parameters or low in nearly lat tidal channels, Journal o Hydraulic Engineering, ACE, 16(). pp Prabhata, K. and Akalank, K.J. (1976) Explicit equation or pipe low problems, Journal o Hydraulics ivision, ACE, Vol., No. 5, pp Ranga Raju, K.G. and Garde, R.J. (1971). iscussion o Flow in conduits with low roughness concentration by John A. Robertson, Journal o Hydraulics ivision, ACE, Vol. 97, No. 1, pp treeter, V.L. (1971). Fluid mechanics, McGraw-Hill, New York, 755p Journal o cience and Technology KNUT ecember 009