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.6 HIGHER ORDER REPRESENTATIONS OF PUMP CURVESThe head produced by a pump has heretofore been defined as a function of the dischargeby fitting a single second-order polynomial through three pairs of points.If the pumpoperation occurs within a relatively narrow discharge range, and these are near the normalcapacity of the pump, then such a simple representation is adequate.When this is not thecase, then more advanced procedures are needed to define well the pump's operatingcharacteristics.Various interpolation procedures can be used for the mathematical represen-tation of a pump curve.This section discusses how pump curves can be duplicated mathe-matically when equations are needed to define their operating characteristics.5.6.1.WITHIN RANGE POLYNOMIAL INTERPOLATIONAny number of values might be used to define a pump characteristic curve, and apolynomial of any order might be used to interpolate the head corresponding to any givendischarge if the range of the discharge values brackets the given discharge.A first-orderpolynomial is simply a straight line.To represent the pump head well with a first-orderpolynomial interpolation, we should first ensure that the smaller discharge Qi is less thanor equal to the given discharge Q, and that the larger discharge Qi+1 is greater than Q.The interpolating function for a first-order polynomial ishp = hpi + h( pi+1 − hpi) Q(− Qi) / Q( i+1 − Qi)(5.32)in which the quantities with subscripts i and i+1 are known, hp is the interpolated headof the pump and Qi ≤ Q ≤ Qi+1.When Q becomes larger than Qi+1, then the first point is dropped and the next point is added.The use of a higher-order polynomial requiresmore data.An n th-order polynomial requires at least n+1 pairs of data points since ann th-order polynomial passes through n+1 points, e.g., a second-order polynomial passesthrough three points, a third-order polynomial through four points etc.The Lagrangeformula is a convenient interpolation formula to use for this purpose because the incrementbetween consecutive values of the independent variable, the discharge Q in this case, neednot be constant.Other formulas do require a constant increment of the independentvariable.The Lagrange interpolation formula isnhp = ∑ FiHi(5.33)i=1in which each Hi is the pump head at point i, and each Fi is the quotient of twoproducts:nnFi = ∏ Q( − Qj) ∏ Q( i − Qj)(5.34)j =1j =1j ≠ ij ≠ iin which the two products Π include n - 1 terms, with the term j = i omitted.Toimplement the Lagrange interpolation successfully in a computer program, tworequirements must be met: (1) the discharge for which the head is wanted must lie within© 2000 by CRC Press LLCthe range of the discharge data points (otherwise the process is extrapolation), and (2) Eqs.5.33 and 5.34 must be properly written.The program LAGRANGE on the CD is designed to read n pairs of points for a pump curve and then provide the pump head forany specified discharge.The program can also be converted into a function subprogramwhich will pass ( hp, Q) pairs to the function from the main program, and an argumentwill specify the Q for which the head is to be determined.Example Problem 5.8A pump curve is shown below.Enter 10 pairs of points from this curve into a file,and then use Lagrange's formula with a third-order polynomial interpolation to obtainvalues of the pump head corresponding to specified discharges, i.e., find hp for dischargesof 850 gal/min, 5800 gal/min, 4200 gal/min, etc.200180160140Pump head, ft12010080100020003000400050006000Discharge, gal/minWe start the solution by selecting 10 discharge values along the abscissa and readingthe corresponding values of pump head to obtain the following:Q800160024003200400045004800520056006000gal/minhp181.5170.0160.0148.5138.6128.0120.4109.095.080.0ft.These data pairs now must be entered into a file that can be read by program LAGRANGE.The input from the keyboard will be 10 2 3, followed by the filename.Then provide thedischarges 850, 5800, 4200, etc.in response to the prompt Give discharge (minusto terminate).The heads returned by the program are the following: Q = 850 gavehp = 181.51, Q = 5800 gave hp = 87.53, Q = 4200 gave hp = 135.16.* * *© 2000 by CRC Press LLC5.6.2.SPLINE FUNCTION INTERPOLATIONOne disadvantage of using Lagrange interpolation is seen when the interpolationinterval shifts to continue to bracket the discharge; then the first derivative, which isneeded in the Newton method, is not continuous.An alternative is to use splineinterpolation.An essential difference between spline and piecewise polynomialinterpolation is that, although a given spline function interpolates only between twoconsecutive points, both the spline function and one or more of its derivatives arecontinuous across these points.We will only discuss cubic splines here, since they requireroughly the same computational effort as quadratic splines and have both continuoussecond and first derivatives across the data points.Cubic splines develop a third-order polynomial between each pair of consecutive pointsas the interpolating function, ory(i) = aix 3 + bix 2 + cix + di(5.35)in which superscript i refers to the segment of the curve before point i, the dependentvariable y plays the role of the pump head hp, and x replaces Q.(For notationalsimplicity let H represent hp in the remainder of this section
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