I am not from Missouri, but I sometimes follow their unofficial motto, "show me". In the study reported here, I had to show myself that the Colebrook-White (CW) formula for pipe friction factors satisfactorily represented data in the transition region between critical flow and fully turbulent flow. I probably could have avoided this exercise by obtaining the original reference by Colebrook [1], but I couldn't find a free, full text version on the web (at the time, but now found). I have used the CW formula in the past without question, mainly because it seemed to be the standard in industry. My recent curiosity about the accuracy of the formula, particularly in the transition region, was prompted by an excellent evaluation of 25 formulas which explicitly predict the friction factor, f, instead of the implicit CW method. This study, by Fred Lusk, compared the formulas to the CW formula. You may find his reports (in Mathcad Prime 3 and in pdf formats) at this link. The attached file documents my process of validation of the CW formula in the transition region. It provides the major references used in the process. For loyal readers of this blog, you may note that I have violated my report guidelines by putting the conclusions last. In my defense, this worksheet is not a report: it is a worksheet that follows the sequence of my investigation. What luck!I found out that the CW formula does model the transition region well for commercial pipes. That is amazing considering the formula was conceived to model the extremes of flow in smooth pipes and fully turbulent flow in rough pipes. The CW formula luckily modeled the intermediate region without any new parameters. As Nikuradse's data showed, the transition behavior of f between those extremes might have been quite different.Alternatives to Colebrook-WhiteThe explicit formulas were needed to simplify the computation of f prior to today's fast computers. Lusk is now timing the various methods using a calculator where speed may still be an issue. Mathcad makes the computation of f from CW almost as good as explicit by using the parameterized nonlinear "Find" routine, as shown by Lusk and in my attached file. The function produced by that process can be used in other equations as if it were f itself. If you are using other math software, even Fortran, you probably can create a function using a root finding algorithm in a similar manner. For a computer program that needs to compute f a huge number of times, the explicit forms may still be an advantage to CW. However, I suspect that some of the methods might actually take longer to compute than CW. One way of estimating the relative speeds is to count the number of mathematical operations. For that count, I suggest that raising a number to a power other than 2 be counted as three operations: (1) log of the base number, (2) multiplication by power, and (3) antilog of product. For the CW operation count, I would assume that the equation needs to be evaluated at least three times. Of course, the time to determine the CW result will depend upon the efficiency of the routine being used. Some other questionsThe absolute roughness for the wrought iron pipes used by Freeman appeared to vary among the pipes. For the amount of variance measured, what is the variance of the friction factor at different flow rates? How does this error compare with the error of the CW formula? The highest relative roughness, k, in the Freeman data was 0.006. Are there data with higher values of k? Do the friction curves for those pipes behave in the same manner as those studied here? Now I have the Colebrook paperI finally found the Colebrook paper, so I didn't need to model the Freeman data. However, I still think the exercise was worthwhile. Also, the Colebrook paper doesn't include any data for pipes with relative roughness greater than 0.006. Back to the web/library! [1] Colebrook, C.F., "Turbulent flow in pipes, with particular reference to the transition region between smooth and rough pipe laws". Journal of the Institution of Civil Engineers (London), (February 1939)
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