Darcy Friction Factor Formulae in Turbulent Pipe Flow


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1 Darcy Friction Factor Formulae in Turbulent Pipe Flow Jukka Kiijärvi Lunowa Fluid Mechanics Paper July 29, 2011 Abstract The Darcy riction actor in turbulent pipe low must be solved rom the Colebrook equation by iteration. Because o the iteration new equations to solve this riction actor has been developed. From these equations the sius, meejain and land equations were validated. All three equations can be used in a smooth pipe, but the meejain and land equations are more accurate than the sius equation. The sius equation can t be used in a rough pipe. Both the meejain and land equation give good results in a rough pipe. 1 Introduction Pressure loss in steady pipe low is calulated using the DarcyWeisbach equation. This equation includes the Darcy riction actor. The exact solution o the Darcy riction actor in turbulent low is got by looking at the Moody diagram [5] or by solving it rom the Colebrook equation [1]. I the Darcy riction actor must be known only once, the Moody diagram is good. This diagram is rather laborious to program in a computer code and doesn t oer any advantages. Unortunately the Colebrook equation must be solved by iteration. Although the solution is ound usually with a ew iterations, the iteration is or example in large pipe low computing codes time consuming and a possible Lunowa is a small company spezialized in technical consulting. Lunowa s webpage 1
2 error source. The iteration needs the irst guess o the value o the Darcy riction actor. How ast the iteration is, depends on how good the irst guess is. Because the boundary layer can be thin in turbulent pipe low, the Darcy riction actor depends on the roughness o the pipe. The Colebrook equation has a term or this roughness. In order to avoid the iteration o the Darcy riction actor rom the Colebrook equation, new equations are deduced. In this study three equations to replace the Colebrook equation are validated. These three are the sius, meejain [6] are land [4] equation. The validation is limited only to steady turbulent pipe low, in which the pipe is assumed to be circular. The solved Darcy riction actors are compared to the solution o the Colebrook equation. In this study the Darcy riction actor solved rom an equation is called according to the person or persons who has developed the equation. For example the meejain riction actor means the Darcy riction actor, which is solved rom the meejain equation. 2 Equations 2.1 DarcyWeisbach The pressure loss in pipe low is calculated using the known DarcyWeisbach equation. This equation is where p L D ρ V 2.2 Colebrook p = L D pressure loss the Darcy riction actor length o pipe or pipe part inner diameter o the pipe density o luid low velocity The Colebrook equation is the most widely used equation to solve the Darcy riction actor. The Colebrook equation is [1] ( 1 e/d = 2.0 log ) Re (2) 2 ρv 2 2 (1)
3 in which e D Re the Darcy riction actor roughness o the pipe inner diameter o the pipe the Reynolds number The ratio e/d is called the relative roughness. The Reynolds number is calculated with the equation Re = V D ν where V is the low velocity and ν the kinematic viscosity o the luid. The Reynolds number tells i the low is laminar or turbulent. I the Reynolds number is smaller than the critical Reynolds number Re cr, the low is laminar. Ater the laminar low regime ollows the transition region. There the low switches between laminar and turbulent randomly. When the Reynolds number reaches a certain value, the low turns rom transitional to turbulent. For pipe low the critical Reynolds number is oten assumed to be The transition region ends approximately at the Reynolds number 4000 [2]. 2.3 sius The sius equation is the most simple equation or solving the Darcy riction actor. Because the sius equation has no term or pipe roughness, it is valid only to smooth pipes. However, the sius equation is sometimes used in rough pipes because o its simplicity. The sius equation is valid up to the Reynolds number 10 5 [3]. The sius equation is (3) = Re 0.25 (4) where means the Darcy riction actor and Re the Reynolds number. 2.4 meejain mee and Jain [6] have devoloped the ollowing equation to the Darcy riction actor [ ( e/d = 0.25 log )] 2 (5) Re 0.9 3
4 in which e D Re the Darcy riction actor roughness o the pipe inner diameter o the pipe the Reynolds number In the meejain equation the Darcy riction actor is solved directly without iteration. 2.5 land land [4] has deduced the equation 1 ( ) 1.11 = 1.8 log e/d (6) 3.7 Re where e D Re the Darcy riction actor roughness o the pipe inner diameter o the pipe the Reynolds number In the land equation there is no need to iterate the Darcy riction actor. The accuracy o the Darcy riction actor solved rom this equation is claimed to be within about ±2 %, i the Reynolds number is greater than 3000 [3]. 3 Code The code or computing the Darcy riction actors was written in Scilab script language. Scilab is a ree sotwarre or numerical computation. It is available in the address The irst guess o the Darcy riction actor in the Colebrook 2 equation is got rom the land equation 6. The right side o the Colebrook equation is solved with this value. Ater that the new riction actor on the let side is calculated. Now the right side o the Colebrook equation 2 is computed with the new Darcy riction actor. Again the let side is solved and a new Darcy riction actor got. The loop is continued until the absolute dierence o the old and new riction actor is low enough. 4
5 Table 1: Computed cases. D pipe diameter, e roughness, e/d relative roughness. 4 Initial values D e e/d mm mm The results were computed with three pipe inner diameters, which were 1.5, 3.0 and 6.0 mm. The pipe was assumed to be drawn tubing. Then the roughness o the pipe is mm [5]. Additionally the Darcy riction actor was computed to a smooth pipe in each diameter. The Reynolds number was between in these calculations. The computed cases are given in the table 1. 5 Results 5.1 Smooth Pipe In the igure 1 are shown the absolute values o the Darcy riction actor solved rom dierent equations. The pipe diameter is 1.5 mm and relative roughness 0. In one region the Darcy riction actor solved rom sius equation is a little higher than the other riction actors. Otherwise the riction actors are almost equal. Because the riction actor in a smooth pipe is only a unction o the Reynolds number, the igures o diameters 3.0 and 6.0 mm are the the same as with the diameter 1.5 mm. In the igure 2 are given the relative errors o the Darcy riction actors solved by dierent equations. The sius riction actor has the greatest relative error 3.5% at the Reynolds number The meejain riction actor has the relative error 2.9 % and the land riction actor 2.5% at the same Reynolds number. Then the sius riction actor has the greatest relative error up to the Reynolds number about The absolute value o the relative error o the sius riction actor increases ater the Reynolds number , 5
6 Col Figure 1: The Darcy riction actor solved rom dierent equations as a unction o the Reynolds number Re. sius equation 4, mee Jain equation 5, land equation 6 and Col Colebrook equation 2. D = 1.5 mm, e = 0, e/d = Figure 2: Relative errors / o the Darcy riction actor solved rom dierent equations as a unction o the Reynolds number Re. sius equation 4, meejain equation 5 and land equation 6. D = 1.5 mm, e = 0, e/d = 0. 6
7 Col Figure 3: The Darcy riction actor solved rom dierent equations as a unction o the Reynolds number Re. sius equation 4, mee Jain equation 5, land equation 6 and Col Colebrook equation 2. D = 1.5 mm, e = mm, e/d = but it is not shown in the igure 2. The relative errors o the meejain and land riction actors are approximately constant ater the Reynolds number Then the relative error o the meejain riction actor is 0.7 % and o the land riction actor 0.9%. 5.2 Rough Pipe The Darcy riction actors solved rom dierent equations are shown in the igures 3, 4 and 5. There can t be seen no signiicant dierence in the riction actors solved rom the meejain, land and Colebrook equation. The sius riction actor diers rom the others. When the relative roughness gets smaller, the sius riction actor diers rom the other riction actors less. The relative errors o the Darcy riction actors solved rom dierent equations are given in the igures 6, 7 and 8. When the Reynolds number is 2300, the relative error gets smaller in every case when the relative roughness decreases. The relative error o the sius riction actor is usually the greatest. At the small Reynolds numbers, the error o the land riction actor is smaller than the error o the mee Jain riction actor. Else the error o the meejain riction actor is the smallest. When the pipe diameter is 6.0 mm, the relative error o the meejain riction actor is only 0.05 % at the Reynolds number
8 Col Figure 4: The Darcy riction actor solved rom dierent equations as a unction o the Reynolds number Re. sius equation 4, mee Jain equation 5, land equation 6 and Col Colebrook equation 2. D = 3.0 mm, e = mm, e/d = Col Figure 5: The Darcy riction actor solved rom dierent equations as a unction o the Reynolds number Re. sius equation 4, mee Jain equation 5, land equation 6 and Col Colebrook equation 2. D = 6.0 mm, e = mm, e/d =
9 Figure 6: The relative error / o the Darcy riction actor solved rom dierent equations as a unction o the Reynolds number Re. sius equation 4, meejain equation 5, land equation 6 and Col Colebrook equation 2. D = 1.5 mm, e = mm, e/d = Figure 7: The relative error / o the Darcy riction actor solved rom dierent equations as a unction o the Reynolds number Re. sius equation 4, meejain equation 5, land equation 6 and Col Colebrook equation 2. D = 3.0 mm, e = mm, e/d =
10 Figure 8: The relative error / o the Darcy riction actor solved rom dierent equations as a unction o the Reynolds number Re. sius equation 4, meejain equation 5, land equation 6 and Col Colebrook equation 2. D = 6.0 mm, e = mm, e/d = Discussion In this study the Darcy riction actor was solved rom our dierent equations in turbulent pipe low. The results were computed in a smooth pipe and with three relative rougnesses o the pipe. The solution o the Colebrook equation was assumed to be the exact solution. The other solved Darcy riction actors were compared to the solution o the Colebrook equation. During the transitional low the greatest relative errors o the solutions o the meejain and land equations were ound. The solution o the sius equation was great in this region, too. There are large uncertainties in the Darcy riction actor in this low regime. In a smooth pipe the solution o the Darcy riction actor rom the sius equation is accurate enough in many applications. In these cases the Reynolds number must be smaller than Both the meejain and land equation are more accurate than the sius equation to solve the Darcy riction actor in a smooth pipe. The sius equation can t replace the Colebrook, meejain and land equation, when the Darcy riction actor must be solved in a rough pipe. The sius equation is appropriate only in a smooth pipe. Both the meejain and land equation give a good approximation o the Darcy riction actor. Based on the computed results it can t be said, 10
11 which one these two equations gives the better solution o the Darcy riction actor. The meejain and land equation can be used in most cases instead o the Colebrook equation to solve the Darcy riction actor. Reerences [1] Colebrook, C., Turbulent Flow in Pipes, with Particular Reerence to the Transition Region between the Smooth and Rough Pipe Laws. Journal o the Institution o Civil Engineers, London, 11, , pp [2] Çengel, Y., Cimbala, J., Fluid Mechanics: Fundamentals and Applications. McGrawHill, 2006, 956 pp. [3] Fox, W., Pritchard, P., McDonald, A., Introduction to Fluid Meachanics. Seventh Edition, John Wiley & Sons, 2010, 754 pp. [4] land, S., Simple and Explicit Formulas or the Friction Factor in Turbulent low. Transactions o ASME, Journal o Fluids Engineering, 103, 1983, pp [5] Moody, L., Friction Factors or Pipe Flow. Transactions o the ASME, 66, 8, November 1944, pp [6] mee, P., Jain, A., Explicit equations or pipelow problems. Journal o the Hydraulics Division (ASCE), 102 (5), 1976, pp
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