Fluid flow in a pipe, described by the pressure drop as a function of flow rate, is commonly modeled with a parabola: ΔP=a*Q2 + b*Q + c. However, it is sometimes very useful to have the inverse relationship: Q(ΔP). In this post we’ll talk about a convenient and accurate way to model that inverse.
This figure demonstrates the normal parabolic model of a typical pipe flow curve. The blue squares show the calculated pressure drop for flow of 40 °F water through a 2 inch diameter pipe, 500 feet long with total minor loss coefficients of 25, and an an elevation increase of 15 feet. The pipe roughness is 0.001 inches. The green circles show three points taken from the total data set which were used to derive a parabola of the form: ΔP=a*Q2 + b*Q + c. Finally, the red line is a plot of the parabola. Clearly, the parabola provides a very good description of the actual pressure-flow relationship, even though it was derived from only three points.
This figure shows what happens if one tries to use an equation of the form: Q=m*(ΔP)2 + n*ΔP + r. As before, the blue squares represent the actual calculated pressure-flow relationship. The green circles are the three points used to derive the parabola. The yellow line is a plot of the parabola. While it passes through the three green circles as required mathematically, it is a poor fit to the data, and is essentially useless beyond 80 gpm.
A better equation to use for Q(ΔP) is suggested by a Taylor series expansion: Q= g*(ΔP)0.5 + k*(ΔP)-0.5 + s. This equation was fit with the same three points (green circles) and is plotted with the purple line. While not quite as good as the parabola representation of the ΔP(Q) relationship, it fits the data reasonably well, and most important, doesn’t veer wildly off as the parabolic Q(ΔP) relationship does.
Next post we’ll talk about application of these models.
Awesome writing.
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