Based on the sales characteristic curves of 201 single stage volute casing pumps, supplied by eight manufacturers, it was determined that the exponent X, by means of the well-known equation NPSH2 = NPSH1(n2 / n1)X, was used to convert NPSH from one speed to another. X always refers to the NPSH-values at the best efficiency point (BEP). In the process of this study, however, the question of which X values exist in the part load and overload range of the Q-NPSH-curves arose.
It was also questioned whether these values are equal as XBEP? These questions were investigated, and the results are presented in this article.
By Jürgen H. Timcke
Three pump companies, denoted as A, B, and F, were arbitrarily selected. Each of these companies manufacture 20 sizes of single stage volute casing pumps, all designed according to the standard EN 733. Three sizes of pump were then arbitrarily chosen, one size from each company, for the investigation. Selected sizes were:
Manufacturer A: 100-80-200 (2/A)
Manufacturer B: 125-100-250 (3/B)
Manufacturer F: 80-65-125 (5/F)
The number in brackets are the short designation of each size to simplify the text and the marking in the diagrams.
Calculated Values of the Exponent X
As the physical connections of NPSH are well-known, this article will not provide an explanation of their characteristics. The primary focus will be instead on the X exponent. More specifically, of interest are only the exponents X in the partload and overload range from Q / Qopt = 0,25 [-] to Q / Qopt = 1,3 [-]. These are calculated from the NPSH-values of the speeds n = 1450 [1/min] and n = 2900 [1/min].
In the figures 2b, 3b and 4b it can be seen that the exponent X is not a constant value. All three sizes show the same behaviour of X = f (Q / Qopt) compared with XBEP:
• X is decreasing in the partload range.
• X is increasing in the overload range.
The consequence of this observation is that each conversion of NPSH-values to lower speeds with a constant value of the exponent X leads to the incorrect NPSH-values, and with it, the incorrect Q-NPSH-curves.
To address this inconsistency, Johann F. Gülich’s Centrifugal Pumps2 describes a method to calculate X-values dependent on NPSH2900 at different values of Q / Qopt.
Sources of Constant Exponent X-values
The sources ,  and  mention constant exponent X-values that should be used for the testing of NPSH. Of all the data that each reference provides, three constant exponents x values were arbitrarily selected to highlight the X value typically used across various sources. The following is a list of direct quotations from each source.
From Reference 4
“… the (NPSH) can be converted by means of the equation (NPSH)T = (NPSH)x(nsp / n)X. The value X = 2 can be used as a first approximation…”
From Reference 6
“NPSHR (Q2, n2) = NPSHR (Q1, n1) x (n2 /n1)² (6). Equation (6) uses the recommendation from the ISO 9906 standard…that an exponent value of 2,0 should be used when testing for NPSH…”
From Reference 7
1. For operating RPM greater than existing: NPSHR2 = NPSHR1 (RPM2 / RPM1)2
2. For operating RPM less than existing: NPSHR2 =
NPSHR1 (RPM2 / RPM1)1.5
All three sources mention the exponent as a constant value: X = 2; X = 1.5 when RPM is less than existing: NPSHR2 = NPSHR1 (RPM2 / RPM1)1.5.
Equation to Calculate Non-constant Exponent X-values
Within Centrifugal Pumps2, Johann F. Gülich explains that, “It would make little sense to apply an exponent of (say) 1.5 at ‘low’ speeds and an exponent of 2.0 at ‘high’ speeds. Rather a continuous function should be used.” Dr. Gülich also highlighted that, “The following tentative formula for scaling the NPSH3 down to lower speeds is suggested… with X = 2 (NPSH3 /NPSHRef)0.3 and NPSHRef = 20m.”
*Note that the procedure is entirely empirical; its only objective is to avoid predicting overly optimistic NPSH3 when scaling down.
The quantity equation X = 2 (NPSH2900 / NPSHRef)0.3, can be very easily converted into a numerical value equation: introduction of the unit ‘m’ for NPSH2900 as well as the numerical value ‘20’ and its unit ‘m’ for NPSHRef. This leads to X = 2 (NPSH2900 [m] / 20 [m])0.3. The unit’s ‘m’ can be cancelled and from them X = 2 (NPSH2900 / 20)0.3 and hence it follows X = 0.814 x NPSH2900 0.3, which simplifies the calculations.
To use a non-constant exponent X to convert NPSH down, is in accordance with the calculation results X = f (Q / Qopt) as shown in the Figures 2b, 3b and 4b. More specifically, to use a constant exponent X, instead of a non-constant one, leads inevitably to incorrect results. In general, this means that the results provide a too low NPSH1450 values, see Figure 7.
NPSH2900 Converted Down to NPSH1450
As an example, the question of how NPSH2900 is converted down to NPSH1450 by changing the value of the X exponent is explored. Specifically, in which way does the value of the exponent X influence the curves NPSH1450 = f (Q), calculated down from the sales characteristic curve NPSH2900 = f (Q)? For this investigation pump size 100-80-200 was used, and the two sales characteristic curves ‘1’ and ‘2’ in Figure 7 were taken over from Figure 2a.
NPSH1450 = f (Q) was calculated with three different values: see lines 5, 7 and 9 in the table of Figure 6. The numerical values of NPSH1450 = f (Q) can be seen in the lines 6, 8 and 10 of Figure 6. These numerical values correspond to the curves C, B and A in Figure 7.
Comparison: Exponent X – curves = f (Q / Qopt)
While in the figures 2a, 3a, and 4a the presented curves X = f (Q / Qopt) are calculated by means of the equation X = lg (NPSH2900 / NPSH1450) / lg 2, it is interesting to compare these curves with those calculated using the equation X = 2 (NPSH2900 / 20)0.3, which also dependent on X = f (Q / Qopt), see Figure 8.
Figure 9 shows an additionally comparison of X = f (Q / Qopt). The curves 2/A, 3/B and 5/F are calculated by means of the equation X = 2 (NPSH2900 / 20)0.3. The straight line of curve 5/F in the partload range until Q / Qopt ≈ 0.6 [-] is based on the straight line of the sales characteristic curve NPSH2900 = f (Q), see Figure 4a. As NPSH = constant in the partload range is theoretically not possible, the calculated curve 5/F in Figure 9 also shows a straight line in the partload range.
Figure 9 also very clearly shows the great numerical difference between X = 2 = const, X = 1.5 = const (based on references ,  and ), and the curves 2/A, 3/B and 5/F with non-constant exponent X-values. This confirms that NPSH2900 calculated down with X = const leads to too low NPSH1450 – curves, as already presented in Figure 7, curves A and B.
By investigating the presented pump sizes and determining the exponent X in the partload and overload range, it has been determined that the X-values are lower and higher, respectively, than the numerical value of XBEP.
Even if in the various technical documentations respectively standards a constant value for the exponent
X is given to convert NPSH to another speed: neither X = 2 nor X = 1.5 are recommendable to calculate NPSH2900 down to NPSH1450. These X-values are not in accordance with the here presented calculation results.
Which recommendation can be given based on the results of this investigation? There is only one:
To convert NPSH2900 down to NPSH1450, or more colloquially to change a high speed to a low one, one must use a non-constant exponent X-value as mentioned in reference 2 and described above.
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