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Units:
bbls/day=barrels/day (1 bbl=42 US gallons), cfm=ft^{3}/min, cfs=ft^{3}/s, cm=centimeter, cP=centipoise, cSt=centistoke, inch H2O=inch water at 60F, inch Hg=inch mercury at 60F, ft=foot, ft H2O= ft water at 60F, g=gram, gpd=gallon (US)/day, gph=gallon (US)/hr, gpm=gallon (US)/min, hp=horsepower, hr=hour, kg=kilogram, km=kilometer, lb=pound, m=meter, mbar=millibar, mm=millimeter, mm Hg=mm mercury at 0C, min=minute, N=Newton, Pa=Pascal (1 Pa=1 N/m^{2}), psf=lb/ft^{2}, psi=lb/inch^{2}, s=second
Topics: Piping Scenarios Equations and Methodology Variables
Minor Loss Coefficients Error Messages References
Introduction
The pump curve pipe flow program automatically intersects a system curve with a pump curve to indicate the
operating point. If you have a pump already installed or want to investigate system
performance of a certain pump before purchasing it, enter two points on its pump
curve along with piping system information to determine the flow rate through the
system. Or, if you know the flow rate 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 Z_{1}Z_{2} and Z_{1}Z_{3}:
If location 1 is above location 2, then Z_{1}Z_{2} should be
entered as positive. If location 2 is above location 1, then Z_{1}Z_{2}
should be entered as negative. Likewise for Z_{1}Z_{3}.
Equations and
Methodology
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The calculation uses the steady state incompressible 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
flow is zero, and flow 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 DarcyWeisbach 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, but numbers are shown with six significant figures.
The calculation does not check if velocities are unreasonable, such as gases flowing at
over half the speed of sound which would require use of compressible gas equations.
Builtin fluid and material properties
You may enter fluid properties or select one of the common liquids or gases
from the dropdown menu. Weight density, kinematic viscosity, and vapor pressure (if
a liquid) for the builtin fluids were obtained from references.
Likewise, enter material roughness or select one of the common
pipe materials listed in the other dropdown menu. Surface roughnesses for the
builtin 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
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Fluid density and viscosity may be entered in a wide choice of units. Some of the
density units are mass density (g/cm^{3}, kg/m^{3}, slug/ft^{3},
lb(mass)/ft^{3}) and some are weight density (N/m^{3}, lb(force)/ft^{3}).
There is no distinction between lb(mass)/ft^{3} and lb(force)/ft^{3}
in the density since they have numerically equivalent values and all densities are
internally converted to N/m^{3}. Likewise, fluid viscosity may be entered in
a wide variety of units. Some of the units are dynamic viscosity (cP, poise, Ns/m^{2}
(same as kg/ms), lb(force)s/ft^{2} (same as slug/fts) and some are kinematic
viscosity (cSt, stoke (same as cm^{2}/s), ft^{2}/s, m^{2}/s).
All viscosities are internally converted to kinematic viscosity in SI units (m^{2}/s).
If necessary, the equation Kinematic viscosity = Dynamic viscosity/Mass density is
used.
A = Pipe inside crosssectional area [L^{2}]
D = Pipe inside diameter [L].
e = Pipe roughness [L].
f = Moody friction factor, used in DarcyWeisbach friction loss equation.
g = Acceleration due to gravity = 32.174 ft/s^{2} = 9.8066 m/s^{2}
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 K_{a}. Only required for liquids.
L = Total pipe length [L].
L_{A} = Length of pipe upstream of pump (pipe A) [L]. Same as L_{a}.
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 P_{atm} and P_{v}, be sure
to enter P_{atm} in the selected units. Only required for liquids.
P_{v} = Vapor pressure of fluid [P]. Expressed as an absolute
pressure. Only required for liquids.
P_{1} = Gage pressure at location 1 of the system [P]. Location 1
could be the surface of a reservoir open to the atmosphere (thus P_{1}=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.
P_{1}P_{3} = Pressure difference between locations 1 and 3 [P].
Q = Flowrate [L^{3}/T]. Also known as discharge or capacity.
Q_{max} = Maximum flowrate on pump curve [L^{3}/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/L^{3}].
Typical units are N/m^{3} or lb(force)/ft^{3}. Note that
S=(mass density)(g)
V_{1} = Velocity of fluid at location 1. This is determined when
a scenario is selected. If location 1 is a reservoir or main (Scenarios B, C, E, and
F), then V_{1} 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 V_{1} is
automatically computed as Q/A.
V_{3} = Velocity of fluid at location 3. This is determined when
a scenario is selected. If location 3 is a reservoir or main (Scenarios B, D, E, and
G), then V_{3} 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 V_{3} is
automatically computed as Q/A.
Z_{1}Z_{2} = 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.
Z_{1}Z_{3} = Elevation of location 1 minus elevation of location
3 [L].
v = Kinematic viscosity of fluid [L^{2}/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 unitless): Back
to Calculations
Compiled from references
Fitting 
K 
Fitting 
K 
Valves: 

Elbows: 

Globe, fully open 
10 
Regular 90^{o}, flanged 
0.3 
Angle, fully open 
2 
Regular 90^{o}, threaded 
1.5 
Gate, fully open 
0.15 
Long radius 90^{o}, flanged 
0.2 
Gate 1/4 closed 
0.26 
Long radius 90^{o}, threaded 
0.7 
Gate, 1/2 closed 
2.1 
Long radius 45^{o}, threaded 
0.2 
Gate, 3/4 closed 
17 
Regular 45^{o}, threaded 
0.4 
Swing check, forward flow 
2 


Swing check, backward flow 
infinity 
Tees: 



Line flow, flanged 
0.2 
180^{o} return bends: 

Line flow, threaded 
0.9 
Flanged 
0.2 
Branch flow, flanged 
1.0 
Threaded 
1.5 
Branch flow, threaded 
2.0 




Pipe Entrance (Reservoir to Pipe): 

Pipe Exit (Pipe to Reservoir) 

Square Connection 
0.5 
Square Connection 
1.0 
Rounded Connection 
0.2 
Rounded Connection 
1.0 
Reentrant (pipe juts into tank) 
1.0 
Reentrant (pipe juts into tank) 
1.0 
Error
Messages
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The following are input checks and will appear if an input is physically impossible, such
as a negative length. Some or all values may be computed even with error messages, even though physically the situation cannot occur.
"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 messages 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.
"P_{1}+P_{atm} must be >0." The sum of P_{1}+P_{atm}
gives P_{1} in absolute pressure. It is physically impossible to
have an absolute pressure ≤ 0 since that implies a complete vacuum at location 1.
"Need P_{1}(P_{1}P_{3})+P_{atm} >0."
The absolute pressure at location 3 is less than 0 absolute. It is physically impossible to have a complete vacuum.
Other messages:
"K must be ≥ 1". If "Q known. Solve for D" and V_{3}=0,
then K must be > 1 in order to solve.
"Tanks open, so P_{1}P_{3}=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 solving for pipe diameter and input data results in negative
losses regardless of pipe diameter, then input data are infeasible.
References
Back to Calculation
Numerical methods citations
Kahaner, D., C. Moler, and S. Nash. 1989. Numerical methods and software.
PrenticeHall, 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. AddisonWesley Publishing Co. 2ed.
Mays, L. W. editor. 1999. Hydraulic design handbook. McGrawHill 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.
PrenticeHall, 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/McGrawHill. 9ed.
White, F. M. 1979. Fluid Mechanics. McGrawHill, Inc.
© 20012022 LMNO Engineering, Research, and
Software, Ltd. All rights reserved.
Please contact us for consulting or other questions about pipe flow.
LMNO Engineering, Research, and Software, Ltd.
7860 Angel Ridge Rd. Athens, Ohio 45701 USA Phone: (740) 7072614
LMNO@LMNOeng.com https://www.LMNOeng.com

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