Pumps & Systems, December 2012
It is common to use graphical techniques to determine the performance of multiple centrifugal pumps operating in series or in parallel. These methods are often time-consuming and rely on the good eyesight of the person making the measurements and a steady hand while drawing the combined performance curves. Numerical techniques can be employed to determine the same overall performance of centrifugal pumps operating in series and in parallel.
|Figure 1. First stage pump performance
||Figure 2. Second stage pump performance
This article will identify the steps for using numerical techniques within an Excel spreadsheet to achieve this goal. It will provide sample calculations of those numerical techniques with detailed cell formulas so that end users can reproduce this technique in their own pump applications. For this article, it is assumed that each pump is a two-stage centrifugal pump that has different performance for the first stage compared to the second stage. These techniques can easily be extended to accommodate multistage pumps that have more than two stages.
|Table 1. Head versus flow arrays for each pump stage
Steps for Numerical Techniques
The individual steps are summarized below:
1. Digitize or make “rough eye” estimates to measure 10 or more (Q, H) points from shutoff to runout conditions for the first stage pump performance curve of head versus flow. Enter the array of first stage head versus flow data into spreadsheet columns.
2. If necessary, use numerical methods to extend the head versus flow data beyond runout. This step is done primarily to prevent subsequent numerical methods from going off into the weeds.
3. Curve fit a polynomial function to the first stage head versus flow data, which may include extrapolated data down to zero head rise.
4. Duplicate steps 1, 2 and 3 with measurements of the second stage head versus flow data. Although it is not essential, it is convenient and recommended that both polynomial curve fits be of the same order.
5. Determine the function of head versus flow for both stages operating together in series. This is done by adding the coefficients of the same ordered terms of the polynomial functions for each stage to establish the applicable coefficients of the polynomial function for the multistage pump.
6. Determine the polynomial function for the combined performance for multiple (and identical) pumps operating in parallel. The set of coefficients for the polynomial will vary depending on the number of pumps running in parallel.
7. Determine the second order polynomial function, which characterizes the resistance (head loss versus flow) of the system piping.
8. Determine the polynomial function that defines overall parallel pump performance minus the system resistance function.
9. Determine the derivative of the polynomial function from the previous step. The derivative function provides the slope of the head versus flow curve, which is needed in the next step.
10. Use iterative numerical methods to solve for head and flow operating points of multiple parallel pumps operating within the specified system. This method uses consecutive rows of spreadsheet calculations to perform successive iterations of the Newton-Raphson method. Results of each iteration eventually converge to the solution of flow rate.