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Introduction

Pipe Network simulates steady flow of liquids or gases under pressure. It can

simulate city water systems, car exhaust manifolds, long pipelines with different

diameter pipes in series, parallel pipes, groundwater flow into a slotted well screen,

soil vapor extraction well design, and more. Enter flows at nodes as positive for inflows and negative for outflows. Inflows plus outflows must sum to 0. Enter one

pressure in the system and all other pressures are computed. All fields must have anumber, but the number can be 0. You do not need to use all the pipes or nodes.

Enter a diameter of 0.0 if a pipe does not exist. If a node is surrounded on all sides by

non-existent pipes, the node's flow must be entered as 0.0. The program allows awide variety of units. After clicking Calculate, the arrows "<--, -->, v, ^" indicate the

direction of flow through each pipe (to the left, right, down, or up).

Losses can be computed by either the Darcy-Weisbach or Hazen-Williams (HW)

method, selectable by clicking on the "Roughness, e" drop-down menu. If HW isused, then the fluid must be selected as "Water, 20C (68F)".

The H,V,Re output field is scrollable using the left and right arrow keys on your

keyboard. Velocity is in m/s if metric units are selected for flowrate Q, and ft/s if

English units are selected for Q.

Equations and Methodology Back to

Calculation

The pipe network calculation uses the steady state energy equation, Darcy Weisbach

or Hazen Williams friction losses, and the Hardy Cross method to determine theflowrate in each pipe, loss in each pipe, and node pressures. Minor losses (due to

valves, pipe bends, etc.) can be accounted for by using the equivalent length of pipe

method.

Hardy Cross Method (Cross, 1936; Viessman and Hammer, 1993) The Hardy Cross method is also known as the single path adjustment method and is a

relaxation method. The flowrate in each pipe is adjusted iteratively until all equations

are satisfied. The method is based on two primary physical laws:

1. The sum of pipe flows into and out of a node equals the flow entering or leavingthe system through the node.2. Hydraulic head (i.e. elevation head + pressure head, Z+P/S) is single-valued. This

means that the hydraulic head at a node is the same whether it is computed from

upstream or downstream directions.

Pipe flows are adjusted iteratively using the following equation,





node elevations Z and known pressure at one node, pressure P at each node is

computed around the network:

P j = S(Z i - Z j - H pipe) + P i where node j is down-gradient from node i. S = fluid

weight density [F/L3].

Minor Losses Minor losses such as pipe elbows, bends, and valves may be included by using the

equivalent length of pipe method (Mays, 1999). Equivalent length (Leq) may becomputed using the following calculator which uses the formula Leq=KD/f. f is the

Darcy-Weisbach friction factor for the pipe containing the fitting, and cannot be

known with certainty until after the pipe network program is run. However, since youneed to know f ahead of time, a reasonable value to use is f=0.02, which is the default

value. We also recommend using f=0.02 even if you select Hazen-Williams losses in

the pipe network calculation. K values are from Mays (1999).

For example, there is a 100-m long 10-cm diameter (inside diameter) pipe with one

fully open gate valve and three regular 90o elbows. Using the minor loss calculator,Leq is 1.0 m and 1.25 m for the fully open gate valve and each elbow, respectively.

The pipe length you should enter into the pipe network calculator is 100 + 1.0 +

3(1.25) = 104.75 m. The calculator allows a variety of units such as m, cm, inch, andft for diameter; and m, km, ft, and miles for equivalent length. If a fitting is not listed,

select "User enters K" and enter the K value for the fitting.

Applications

The pipe network calculation has many applications. Two examples will be provided.

1. Municipal water supply system. A water tower is located at node D. The other

nodes could represent industries or homes. Enter the water withdrawals at all the

nodes as negative numbers, then enter the inflow to the network from the water tower at node D as a positive number equal to the sum of the withdrawals from the other

nodes. Usually, cities require a certain minimum pressure everywhere in the system,

often 40 psi. Use the drop-down menu to select the node that you expect will have thelowest pressure - possibly the node furthest from D or the one at the highest elevation;

we'll use node I. Enter the pressure at node I as 40 psi. Enter all the pipe lengths,

diameters and node elevations. Then click "Calculate". You can use your right andleft arrow keys to scroll to the left and right to see the velocity in each pipe.

Typically, you want pipe velocities to be around 2 ft/s. If you are designing a system

(as opposed to analyzing a system that is already in place), vary the pipe diameters



until the pipe velocities are reasonable and pressure at node D is as low as possible to

minimize the height of the water tower. There will be a trade-off between pressure atD and pipe diameters. Smaller diameter pipes will save money on pipes but will

require a taller water tower. The water tower height is proportional to the pressure at

D according to h=P/S, where P is the pressure at D. S is the weight density of the

water, and h is the water tower height required. A more detailed example.

2. Manifold. A manifold has multiple inflows at various positions along the same

pipeline, and one outflow. Let node I be the outflow and use all other nodes A-H as

inflow locations; so flow is from node A through pipes 1, 2, 5, 7, 6, 8, 11, and 12 and

out node I. Enter the diameters and lengths of these pipes and the desired inflows at

nodes A-H. Enter the outflow at node I as a positive number equal to the sum of the

inflows at nodes A-H. Enter the diameters of pipes 3, 4, 9, and 10 as 0.0 since they

are non-existent pipes. Enter the elevations of all nodes. For a horizontal pipe, set allthe elevations to the same value or just to 0.0 to keep it simple. From the drop-down

menu, select the node where you know the pressure and enter its pressure. Clicking

"Calculate" will give the flowrate in all pipes and the pressure at all the nodes.

Built-in fluid and material propertiesThe user may manually enter fluid density and viscosity or select one of the common

liquids or gases from the drop-down menu. Density and viscosity for the built-influids were obtained from Munson et al. (1998). Likewise, the user may manually

enter material roughness or Hazen-Williams C, 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 Munson et al. (1998), Streeter et al. (1998) and Mays(1999).

Units

bbls/day=barrels/day, cfm=ft3/min, cfs=ft3/s, cm=centimeter, cP=centipoise,

cSt=centistoke, in=inch, in H2O=inch water at 60F, in Hg=inch mercury at 60F,

ft=foot, g=gram, gpd=gallon (US)/day, gph=gallon (US)/hr, gpm=gallon (US)/min,hr=hour, kg=kilogram, km=kilometer, lb=pound, lb(f)=pound (force), m=meter,

mbar=millibar, mm=millimeter, mm H2O=mm water at 4C, min=minute, N=Newton,

psi=lb(f)/in2, s=second

Variables [] indicates units: F=force, L=length, P=pressure,

T=time Back to Calculation

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 areweight 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 ina 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 equationKinematic viscosity = Dynamic viscosity/Mass density is used internally.

A = Pipe area [L2].

C = Hazen Williams coefficient. Selectable as last item in drop-down menu saying

"Roughness, e".

D = Pipe diameter [L].

e = Pipe roughness [L]. All pipes must have the same roughness.

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 = Head losses in pipe [L]. Can also be expressed in pressure units [P].

k = Constant in Hazen Williams equation for computing H .K = Minor loss coefficient.

L = Pipe length [L].

Leq = Equivalent length of pipe for minor losses [L].n = Constant used in Hardy Cross equation.

P = Node pressure [P]. Can also be expressed in length units [L].

Q = Flowrate through pipe, or into or out of node [L3/T]. Also known as discharge or capacity.

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)

V = Velocity in pipe [L/T].

Z = Elevation of node [L].Z+P/ S = Hydraulic head [L]. Also known as piezometric head. Can also be

expressed in pressure units [P].

v = Kinematic viscosity of fluid [L2/T]. Greek letter "nu". Note that kinematicviscosity is equivalent to dynamic (or absolute) viscosity divided by mass density.

Mass density=S/g.

Error Messages in Pipe Network calculation Back to Calculation

"N ode Q' s must sum to 0" . Check the node flowrates that you entered. Total flow

into pipe network must equal total flow out of pipe network.

"T otal inflow must be >0" . Check that you have positive flow into the system. Youhave entered all node flows as 0.0 or negative.

"N ode i must have Q=0" . Node i is completely surrounded by pipes having diameters



less than 0.001 m, which is the criteria the program uses for treating pipes as being

non-existent. You cannot have flow in or out of a node that is surrounded by non-existent pipes.

"|Q| must be < 1e9 m3/ s" . Node flows cannot exceed 109 m3/s. | | is absolute value.

" P at isolated node." Be sure that the "P known at node x" drop-down menu indicates

a node that is surrounded by at least one existing pipe (i.e. a pipe having a diameter greater than 0.001 m). If you don't know the pressure anywhere in your system, just

enter 0.0 for the pressure. All the other node pressures will be computed relative tothe pressure you enter.

" Density must be > 0" , " Density too high" , " Viscosity must be > 0" , " Viscosity too

high." . These messages can only occur if "Another fluid" is selected from the fluid

drop-down menu. Be sure the density and viscosity you enter are greater than zerobut less than 1010 kg/m3 and 1010 m2/s, respectively.

" D must be < 1e6 m" . Individual pipe diameters cannot exceed 106 m.

" L must be < 1e7 m" . Individual pipe lengths cannot exceed 107 m."|Z | must be < 1e20" , " |P | must be < 1e20 m" . The absolute value of each node

elevation and pressure that are input cannot exceed 1020 m.

"N eed Water (20C ) if H-W " . If "Hazen-Williams C" is selected from the Roughnessdrop-down menu, you must also select "Water, 20C (68F)" from the fluid drop-down

menu. The Hazen-Williams method for head losses is only valid for water at typical

city water supply temperatures, such as 20C.

" C out of range" , " e out of range" . These messages can only occur if you selected

"Another material" from the pipe material drop-down menu. Valid ranges are

0<C<1000 and 0 <= e < 10.0 m. Normally, C will not exceed 150 and e will not

exceed 0.001 m, but we allow high ranges for those who like to experiment." P ipe i e/ D out of range" . See the equations above for Friction loss computation

using Darcy-Weisbach. e/D cannot exceed 0.05 unless Reynolds number is less than4000. Also, e/D cannot be 0.0 (i.e. e cannot be 0.0) if Reynolds number is greater

than 108.

"U nusual input." If you experiment with the calculation long enough, you may enter some very unusual input combinations. Some situations are physically not possible,

but the calculation will continue iterating to compute the pipe flows and losses. After

5000 iterations (a few seconds of real time), the program will stop running and give

you this error message, so you can check your input and enter more realistic numbers.

The program has been designed so that it will not "lock up".Other things. If the calculation doesn't seem to run when you click "Calculate", check

your inputs. If you accidentally entered two decimal points or a letter in an input

field, then it won't run and won't give an error message.

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