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Topics:  Piping Scenarios   Equations and Methodology   Variables   Minor Loss Coefficients   Error Messages  References

Introduction
This program automatically intersects a system curve with a pump curve to tell you the operating point.  If you have a pump already installed or want to investigate system performance of a certain pump before purchasing it, you can enter two points on its pump curve along with piping system information to determine the actual flowrate through the system.  Or, if you know the flowrate or velocity, you can solve for diameter, pipe length, pressure difference, elevation difference, or the sum of the minor loss coefficients.

A pump curve (blower curve for gases) is incorporated into the calculation to simulate systems containing a centrifugal pump or other pump that has a pump curve.  To keep the calculation's input relatively simple, we only require you to enter two points on the pump curve - flow at zero head and head at zero flow.  A parabolic curve is then formed between the two points as shown in equations below.  The calculation also asks for information specifically about the pipe on the suction side of the pump.  This information is used to compute the net positive suction head available (NPSH_A) for liquids.  For a pump to properly function, the NPSH_A must be greater than the NPSH required by the pump (obtained from the pump manufacturer).  If your system does not require a pump or uses a pump that does not have a parabolically shaped pump curve, then our other Darcy Weisbach design calculation may be more helpful.

Piping Scenarios
Pipe A is the pipe upstream from the pump (i.e. the suction side pipe).
Convention for Z₁-Z₂ and Z₁-Z₃:  If location 1 is above location 2, then Z₁-Z₂ should be entered as positive.  If location 2 is above location 1, then Z₁-Z₂ should be entered as negative.  Likewise for Z₁-Z₃.

Equations and Methodology                                    Back to Calculations
The calculation on this page uses the steady state energy equation.  Minor losses (due to valves, pipe bends, etc.) and major losses (due to pipe friction) are included.   The Darcy Weisbach equation for friction losses is used and the calculation includes both laminar and turbulent flow.  The equations are standard equations which can be found in most fluid mechanics textbooks (see references below).  A pump curve is included in the calculation.  Determination of the pump curve requires that the user enter the two extreme points on the curve - head when capacity is zero, and capacity when head is zero.  Then, a parabola with a negative curvature is fit through the two points.  This parabola is used since it is a good approximation of a typical pump curve and does not require users to enter a multitude of data points.  And, oftentimes, pump catalogs only give the two extreme points on the curve rather than a graph showing the complete curve.

Energy equation with Darcy-Weisbach friction losses
All equations were compiled from references except for parabolic pump curve equation which is our development.  The Colebrook equation is an equation representation of the Moody diagram.

Pump Curve
To provide an example of a pump curve developed using the equation H=H_max[1-(Q/Q_max)²], let Q_max=1500 gpm (when head is zero) and H_max=900 ft (when Q is zero).  The pump curve used in the calculation will look like:

The Colebrook equation is solved for f using Newton's method (Kahaner et al., 1989).  The remaining calculations are analytic (i.e. closed form) except "Solve for V, Q", "Q known. Solve for Diameter", and "V known. Solve for Diameter".  These three calculations required a numerical solution.  Our solution utilizes a cubic solver (Rao, 1985) with the result accurate to 8 significant digits.  Multiple solutions are possible for the three numerical solutions.  All solutions for both laminar and turbulent flow are automatically determined and shown, if they exist.  All of the calculations utilize double precision.

Built-in fluid and material properties
The user may enter his own fluid properties or select one of the common liquids or gases from the drop-down menu.  Weight density, kinematic viscosity, and vapor pressure (if a liquid) for the built-in fluids were obtained from references.   Likewise, the user may enter his own material roughness or select one of the common pipe materials listed in the other drop-down menu.  Surface roughnesses for the built-in materials were compiled from references.

Net Positive Suction Head
NPSH is the sum of the heads that push fluid into a pump less the suction side losses.  Most pumps have a minimum requirement for NPSH, called NPSH_R.  If the NPSH available by the piping system (NPSH_A) is lower than NPSH_R, then the pump will not function properly and may overheat.   NPSH is only defined for liquids.

Variables   Units:  F=force, L=length, P=pressure, T=time                    Back to Calculations

Fluid density and viscosity may be entered in a wide choice of units.  Some of the density units are mass density (g/cm³, kg/m³, slug/ft³, lb(mass)/ft³) and some are weight density (N/m³, lb(force)/ft³).   There is no distinction between lb(mass)/ft³ and lb(force)/ft³ in the density since they have numerically equivalent values and all densities are internally converted to N/m³.  Likewise, fluid viscosity may be entered in a wide variety of units.  Some of the units are dynamic viscosity (cP, poise, N-s/m2 (same as kg/m-s), lb(force)-s/ft² (same as slug/ft-s) and some are kinematic viscosity (cSt, stoke (same as cm²/s), ft²/s, m²/s).   All viscosities are internally converted to kinematic viscosity in SI units (m²/s).   If necessary, the equation Kinematic viscosity = Dynamic viscosity/Mass density is used.

A = Pipe area [L²]
D = Pipe diameter [L].
e = Pipe roughness [L].
f = Moody friction factor, used in Darcy-Weisbach friction loss equation.
g = Acceleration due to gravity = 32.174 ft/s² = 9.8066 m/s²
h_f = Major losses for entire pipe [L].  Also known as friction losses.
h_fA = Major losses for pipe upstream of pump (pipe A) only [L].
h_m = Minor losses for entire pipe [L].
h_mA = Minor losses for pipe upstream of pump (pipe A) only [L].
H = Total dynamic head [L].  Also known as system head or head supplied by pump.
H_max = Maximum head that pump can provide [L].  It is the head when Q=0.
K = Sum of minor loss coefficients for entire pipe.  See table below for values.
K_A = Sum of minor loss coefficients for pipe upstream of pump (pipe A).  Same as Ka.  Only required for liquids.
L = Total pipe length [L].
L_A = Length of pipe upstream of pump (pipe A) [L].  Same as La.   Only required for liquids.
NPSH = Net positive suction head [L].  The calculation computes NPSH_A (NPSH available).
P_atm = Atmospheric (or barometric) pressure [P].  Standard atmospheric pressure = 14.7 psi = 29.92 inch Hg = 760 mm Hg = 1 atm = 101,325 Pa = 1.01 bar.  Note that your local atmospheric pressure is different from standard atmospheric pressure.  Be careful - if you change the units of Patm and Pv, be sure to enter Patm in the selected units.  Only required for liquids.
P_v = Vapor pressure of fluid [P].  Expressed as an absolute pressure.  Only required for liquids.
P₁ = Gage pressure at location 1 of the system [P].  Location 1 could be the surface of a reservoir open to the atmosphere (thus P₁=0), or the pressure in a supply main (same as a tank under pressure), or location 1 could simply be a location in a pipe upstream of the pump.  Only required for liquids.
P₁-P₃ = Pressure difference between locations 1 and 3 [P].
Q = Flowrate [L³/T].  Also known as discharge or capacity.
Q_max = Maximum flowrate on pump curve [L³/T].  Corresponds to point on pump curve where head is zero.
Re = Reynolds number.
S = Specific Weight of Fluid (i.e. weight density; weight per unit volume) [F/L³].   Typical units are N/m³ or lb(force)/ft³.  Note that S=(mass density)(g)
V₁ = Velocity of fluid at location 1.  This is determined when you select a scenario.   If location 1 is a reservoir or main (Scenarios B, C, E, and F), then V₁ is automatically set to 0 because the velocity head of the fluid in the reservoir or main (or pressure tank) is much smaller than in the attached pipeline.  This is a standard assumption in fluid mechanics.  However, if location 1 is inside the suction side pipeline, then V₁ is automatically computed as Q/A.
V₃ = Velocity of fluid at location 3.  This is determined when you select a scenario.   If location 3 is a reservoir or main (Scenarios B, D, E, and G), then V₃ is automatically set to 0 because the velocity head of the fluid in the reservoir or main (or pressure tank) is much smaller than in the attached pipeline.  This is a standard assumption in fluid mechanics.  However, if location 3 is inside your discharge side pipeline, then V₃ is automatically computed as Q/A.
Z₁-Z₂ = Elevation of location 1 minus elevation of pump [L].  If the pump is above location 1, then enter this value as negative.  Only required for liquids.
Z₁-Z₃ = Elevation of location 1 minus elevation of location 3 [L].
v = Kinematic viscosity of fluid [L²/T].  greek letter "nu".  Note that kinematic viscosity is equivalent to dynamic (or absolute) viscosity divided by mass density.  Mass density=S/g.

Table of Minor Loss Coefficients (K is unit-less):        Back to Calculations
Compiled from references

Error Messages                         Back to Calculations
The following are input checks and will appear if an input is physically impossible, such as a negative length.:
"Q, V, D, L must be > 0", "Density, Viscosity must be > 0", "K must be >= 0", "e must be >= 0", "Q_max , H_max must be > 0", "Q must be <= Q_max".

The following are input checks for liquids only:
"L_a, K_a must be >= 0", "Vapor and Atm P must be > 0".
"Need L_a<=L and K_a<=K".  Length of the suction pipe (Pipe A) was entered as being longer than all of the pipe, or K for the suction pipe was entered as greater than K for the entire system.

Other messages:
"K must be >=1".  If "Q known. Solve for D" and V₃=0, then K must be > 1 in order to solve.
"Tanks open, so P₁-P₃=0 for B".  Cannot solve for pressure difference if using Scenario B since reservoirs are defined to be at zero pressure thus zero pressure difference.

"Infeasible input. H<0", "Infeasible input. h_m<0", "Infeasible input. h_f <=0", "Re or e/D out of range", "Infeasible. Losses will be <=0", "f won't be 0.008 to 0.1", "f will be too small", "f will be too large", "Re will be > 1e8", "Infeasible input".  One of these messages will appear if each of your inputs is okay, but they combine to give no possible solution.  For instance, if you are solving for pipe diameter and your input data will result in negative losses regardless of pipe diameter, then your data are infeasible.

References                                          Back to Calculations
Numerical methods citations
Kahaner, D., C. Moler, S. Nash.  1989.  Numerical methods and software.  Prentice-Hall, Inc.

Rao, S. S.  1985.  Optimization theory and applications.  Wiley Eastern Limited.  2ed.

Fluid mechanics references
Gerhart, P. M, R. J. Gross, and J. I. Hochstein.  1992.  Fundamentals of Fluid Mechanics.  Addison-Wesley Pubishing Co.  2ed.

Mays, L. W. editor.  1999.  Hydraulic design handbook.  McGraw-Hill Book Co.

Munson, B.R., D. F. Young, and T. H. Okiishi.  1998.  Fundamentals of Fluid Mechanics.  John Wiley and Sons, Inc.  3ed.

Potter, M. C. and D. C. Wiggert.  1991.  Mechanics of Fluids.   Prentice-Hall, Inc.

Roberson, J. A. and C. T. Crowe.  1990.  Engineering Fluid Mechanics.   Houghton Mifflin Co.

Streeter, V. L., E. B. Wylie, and K. W. Bedford.  1998.  Fluid Mechanics.   WCB/McGraw-Hill. 9ed.

White, F. M.  1979.  Fluid Mechanics.  McGraw-Hill, Inc.

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