Pump and Fan Performance: Method

Pump and Fan Performance: Curve Fit, System Curve, Affinity Laws, Specific Speed, NPSH

Scope

The tool answers the questions an engineer asks of a manufacturer's performance curve: where will this centrifugal pump or fan run in my system, how much power will it draw and at what efficiency, what happens if I change its speed or trim its impeller, will two of them in parallel or in series do what I need, and, for a pump, will it cavitate. It takes the curve as pasted points (flow and head or total pressure, with efficiency and shaft power when the chart gives them), fits a smooth polynomial through them, builds the system curve from the static head and one loss term, and works everything from the intersection. It is a performance tool: it prints results, not margins of safety. The one comparison it makes is the net positive suction head available against the pump's requirement, printed as a plain difference and ratio.

Conventions and units

Flows, heads, pressures, densities, speeds and diameters go through the site's unit dictionary (gpm, ft, psi, lb/ft³, rpm, in and their SI counterparts). The pasted curve keeps its own units, chosen in the "Curve Units" select, and is converted once; those units are not changed by the unit-system switch. A pump curve is given as head of the pumped liquid, which is independent of density: the same pump produces the same head of oil as of water, but a different pressure and a different power. A fan curve is given as total pressure at a rated air density, so the fan's pressure and power scale with the actual density and the tool corrects them before anything else. Efficiencies are entered in percent (a column whose values never exceed 1.5 is read as fractions); the shaft power column takes the unit of the "Power Column" select. All results are computed in SI and shown in the selected system.

Fitting the curve

A pasted curve is a handful of points read from a chart, so the tool fits a least-squares polynomial through them, quadratic by default, cubic on request: H(Q) = c₀ + c₁ Q + c₂ Q² (+ c₃ Q³). The fit is done on the flow scaled by its largest value so that the normal equations are well conditioned, and the coefficients are printed in the display units and in the pasted units. The table of pasted against fitted values, the R², the rms and the largest residual tell how well the polynomial follows the chart: a residual of a few tenths of a percent of the head is the precision of a chart reading; a larger one means the curve has a shape the polynomial cannot follow (a drooping or unstable curve near shut-off, an axial-flow fan's stall dip), and a cubic, or a curve pasted only over the range of interest, will do better. The efficiency and power columns are fitted with polynomials of the same order; the best-efficiency point (BEP) is the maximum of the fitted efficiency curve. A quadratic through an efficiency curve that peaks is a good description near the peak and worse toward shut-off and run-out; the tool uses it for the operating-point efficiency and for the BEP, both near the peak in a sensible selection.

The system curve

What the system demands of the machine is the static head, the elevation (or pressure) difference between the two liquid surfaces that does not depend on the flow, plus the friction and fitting losses, which for turbulent flow rise nearly with the square of the flow: H_sys(Q) = H_st + K Q². The tool fixes K from one design point, the head the system needs at the design flow (from the Pipe Flow or Duct Sizing tool, or a hydraulic calculation), K = (H_d − H_st)/Q_d²; or K is entered directly in the pasted curve's units. The parabola is a simplification: the friction factor falls slowly with Reynolds number, so the true exponent is a little below 2 (the Pipe Flow tool recomputes Colebrook at every flow), and a system with a control valve has a K that changes with the valve position. For a fan system the static term is usually zero and the whole curve is K Q².

The operating point

The machine runs where its curve crosses the system curve: that flow is the only one at which the head the machine produces equals the head the system absorbs. The tool finds it by scanning the fitted curve for a sign change of H(Q) − H_sys(Q) and bisecting to machine precision; a crossing up to 15 % beyond the last pasted point is accepted and flagged as extrapolated, further than that the tool reports no operating point (a static head above the shut-off head, or a curve pasted over too short a range). At the operating point the fluid power is ρ g Q H for a pump (Q Δp for a fan), the shaft or brake power is the fluid power divided by the efficiency read from the fitted efficiency curve, or the fitted power column, or the entered efficiency when the curve has neither. The Hydraulic Institute's US form P = Q H s / 3960 hp (gpm, ft, specific gravity) is the same expression with water at 62.3 lb/ft³ folded into the constant. The operating flow as a fraction of the BEP flow says how well the pump is matched: most manufacturers want continuous operation between about 70 and 120 % of BEP flow, where the efficiency is near its peak and the radial thrust, vibration and recirculation are low.

Specific speed

Specific speed is the index by which impeller types are compared and by which a pump is placed on its design chart: n_s = N √Q / H^3/4, taken at the best-efficiency point with the full impeller. ANSI/HI 14.1-14.2 defines the metric form with N in rpm, Q in m³/s and H in m, and the US form with Q in gpm and H in ft, and gives 51.6 as the factor from metric to US (the exact value from the gallon and the foot is 51.645); a form with Q in m³/h is 60 times the metric value. The standard also defines the dimensionless type number K = ω √Q / (g H)^3/4 (ω in rad/s), with n_s = 52.92 K and N_s = 2,733.72 K. Radial-flow impellers lie below about N_s = 1,500 (US), mixed-flow between 1,500 and about 7,000, axial-flow above; at a given speed a low specific speed means a high head and a small flow, the domain of the narrow radial impeller. For a fan the tool prints the US form with cfm and inches of water at the actual density, and the type number with the specific energy Δp/ρ in place of g H. The suction specific speed S = N √Q / NPSH_r^3/4 uses the same units; the standard puts typical inlet designs between 6,000 and 13,000 (120 to 250 metric), and higher values, obtained with large impeller eyes, come with a narrower stable flow range.

Net positive suction head

A pump cavitates when the pressure at the impeller inlet falls to the vapor pressure of the liquid. The margin against that is the net positive suction head available, the absolute total head at the impeller datum above the vapor-pressure head (ANSI/HI 14.1-14.2 §14.2.5.5): NPSH_a = (p_s − p_v)/(ρ g) + z_s − h_fs, with p_s the absolute pressure on the source surface (atmospheric for an open tank, lower at altitude), p_v the vapor pressure at the pumping temperature, z_s the height of the source surface above the impeller centreline (positive for a flooded suction, negative for a lift) and h_fs the losses of the suction line at the operating flow. The manufacturer's NPSH required, NPSH₃, is the value at which the first-stage head has already dropped 3 % through cavitation; the pump needs more than that. ANSI/HI 9.6.1 sets the margin by the service: a ratio NPSH_a/NPSH_r of 1.1 to 1.3, or a fixed 0.6 m (2 ft), is common for ordinary water pumps, with larger ratios for high-energy pumps, hot or volatile liquids and pumps that must run far from their BEP. The tool prints the two values, their difference and ratio, and says whether the available head exceeds the required one; it does not compute a margin of safety. The NPSH required rises with the square of the speed (approximately) and with flow, so a speed increase or an operating point to the right of BEP needs the check repeated.

The affinity laws

Two geometrically similar machines running at the same flow coefficient Q/(N D³) have the same head coefficient g H/(N² D²) and the same power coefficient P/(ρ N³ D⁵). Applied to one machine at two speeds, or to two impellers of different diameter in the same casing, these are the affinity laws: Q₂/Q₁ = (N₂/N₁)(D₂/D₁)³, H₂/H₁ = (N₂/N₁)²(D₂/D₁)², P₂/P₁ = (N₂/N₁)³(D₂/D₁)⁵, with the efficiency unchanged along each homologous line (the SCS irrigation handbook states them as items 7 and 8 of its pump characteristics, with its Figure 8-2 of capacity, head and brake horsepower against speed in percent). For a fan the head law and the power law carry the density ratio as well. The speed laws are exact for similar flow and very good in practice over a 2:1 range, the small changes in Reynolds number and in relative clearance apart. The diameter laws for an impeller trimmed in its own casing are approximate: a trim changes the outlet width and the casing match, not the whole geometry, so manufacturers' trim curves fall a little below the D² head law and the usual practice (Karassik's Pump Handbook, the SCS handbook's item 8, the Engineering ToolBox example) scales the flow with D and the power with D³, not with D³ and D⁵ as the full similarity laws do for a machine of another size. The "Diameter Law" select chooses between the two; the trim form is the default because a new diameter entered for the same machine is nearly always a trim. Keep trims under about 20 % and read the printed factors as estimates.

Scaling a fitted curve is algebraic: H₂(Q) = k_H H₁(Q/k_Q), so each coefficient of Qⁱ is multiplied by k_H/k_Qⁱ, and η₂(Q) = η₁(Q/k_Q). The tool prints the scaled curve, the new operating point on the unchanged system curve and, separately, the homologous point: the old operating point moved by the three factors. The two are the same only when the static head is zero, because homologous points lie on a parabola through the origin while the system curve starts at H_st. With a static head, slowing a pump cuts the flow less than the speed ratio and the power less than its cube; this is why a variable-speed drive saves most on systems with little static head. The tool also solves for the speed that puts the curve exactly through the design point, s² H₁(Q_d/s) = H_d, the usual question when a pump is slightly oversized.

Two machines in parallel or in series

Two identical machines in parallel deliver twice the flow at each head, H_2p(Q) = H₁(Q/2); in series they add their heads at each flow, H_2s(Q) = 2 H₁(Q). The combined curve is crossed with the same system curve, and each machine runs at its share, half the flow at the full head in parallel, the full flow at half the head in series, with the efficiency of the single-machine curve at that share and the total shaft power twice that of one. Because the system curve rises with Q², two pumps in parallel deliver well under twice the flow of one (in the default case 1.14 times), and each runs to the left of its single-machine operating point, at a lower efficiency; series pumping suits a system dominated by static head, parallel pumping one dominated by friction. The tool does not model dissimilar machines or a check valve closing on the weaker one.

Fans

Everything above applies to a centrifugal or axial fan with the head replaced by the total pressure Δp and the fluid power by Q Δp. Fan curves are rated at standard air (0.075 lb/ft³, 1.2 kg/m³); at another density the pressure and power scale with ρ_a/ρ_r and the volume flow does not (the fan laws at constant speed and size), so a fan selected at sea level for a plant at altitude delivers the same cfm against a lower pressure. The tool corrects the pasted curve before the fit, and the Fan Operating Point tool of the HVAC area, which takes a fan curve against a duct system at two speeds, is the simpler companion for that case. The common air-horsepower constant, hp = cfm × in. w.g. / 6,356, is the fluid power with 5.2 lbf/ft² per inch of water; the tool uses the dictionary's inch of water at 4 °C (249.082 Pa), which gives 6,343.5, a 0.2 % difference.

Reading the results

Check the fit table first: if the residuals are a small fraction of the head, the operating point is as good as the chart reading; if not, the fit is the problem, not the pump. The operating flow against the BEP flow and the specific speed tell whether the right pump has been chosen; the efficiency and shaft power size the motor (add the usual service factor and the motor's own efficiency); the NPSH comparison decides whether the suction arrangement works. The scaled results show what a speed change or trim buys, and the speed through the design point is the setting a variable-speed drive would take. The default case is illustrative, a textbook-style pump against a 40 ft static head, and the charts carry every curve and point so the geometry of the match can be seen at once.

Limits

Polynomial fit of the pasted points (no extrapolation beyond 15 % of the last point); a system curve of the form H_st + K Q² with constant K; steady operation of one machine or two identical ones; the affinity laws as stated, with the efficiency held constant and the diameter law only for geometric similarity; NPSH at the impeller datum with the suction losses entered, the NPSH required taken as entered (its own flow and speed dependence is not modelled); no viscosity correction of the pump curve (Hydraulic Institute 9.6.7 for liquids above about 20 cSt), no transient, surge or water-hammer effects, no motor or drive efficiency.