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UNIVERSITY OF ILLINOIS BULLETIN
IsSUED WEEKLY
Vol. XXXIV November 13, 1936 No. 22
[Entered as second-class matter December 11, 1912, at the post office at Urbana, Illinois, under
the Act of August 24, 1912. Acceptance for mailng at the secial rate of postage provided
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ANALYSIS OF FLOW IN NETWORKS OF
CONDUITS OR CONDUCTORS
BY
HARDY CROSS
BULLETIN No. 286
ENGINEERING EXPERIMENT STATION
PUBLIsHED Br TH UMNIVRSIT OF ILLINOIS, URBANA
Puca: Tranr-rnu COrra
T - HE Engineering Experiment Station was established by act
of the Board of Trustees of the University of Illinois on De-
cember 8, 1903. It is the purpose of the Station to conduct
investigations and make studies of importance to the engineering,
manufacturing, railway, mining, and other industrial interests of the
State.
The management of the Engineering Experiment Station is vested
in an Executive Staff composed of the Director and his Assistant, the
Heads of the several Departments in the College of Engineering, and
the Professor of Industrial Chemistry. This Staff is responsible for
the establishment of general policies governing the work of the Station,
including the approval of material for. publication. All members of
the teaching staff of the College are encouraged to engage in scientific
research, either directly or in co6peration with the Research Corps
composed of full-time research assistants, research graduate assistants,
and special investigators.
To render the results of its scientific investigations available to
the public, the Engineering Experiment Station publishes and dis-
tributes a series of bulletins. Occasionally it publishes circulars of
timely. interest, presenting information of importance, compiled from
various sources which may not readily be accessible to the clientele
of the Station, and reprints of articles appearing in the technical press
written by members of the staff.
The volume and number at the top of the front cover page: are
merely arbitrary numbers and refer tothe general publications of the
University. Either above the title or below the seal is given the num-
ber of the Engineering Experiment Station bulletin, circular, or reprint
which should be used in referring to these publications.
For copies of publications or for other information address
THE ENGIN'EERING EXPERIMENT STATION,,
SUNIVERSITY OF ILLINOIS,
URBANA, ILLINOIS
UNIVERSITY OF ILLINOIS
ENGINEERING EXPERIMENT STATION
BULLETIN No. 286
NOVEMBER, 1936
ANALYSIS OF FLOW IN NETWORKS OF
CONDUITS OR CONDUCTORS
BY
HARDY CROSS
PROFESSOR OF STRUCTURAL ENGINEERING
ENGINEERING EXPERIMENT STATION
PUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA
UNIVERSIIY
OF ILLINOIS
4000-7-36-10216 6 - 'NIOR
CONTENTS
I. INTRODUCTION . . . . . . . .
1. Type of Problem. . . . . .
2. Flow of Water in a Network of Pipes
II. METHODS OF ANALYSIS . . . .
3. Methods of Analysis Proposed
PAGE
7
7
8
* . . . 8
. . . . 8
III. METHOD OF BALANCING HEADS . . . . . . . .
4. Statement of Method
5. Proof of Method . . . . . . . . . . .
6. Illustrative Problems
Problem 1. Single Closed Circuit
(a) h varies as Q . . .
(b) h varies as Q
(c) h varies as Q2
Problem 2. Simple Network-h varies as Q2
Problem 3. Complex Network-Two Inlets-h
varies as Q
Problem 4. Systems of Pipes in Different Planes
Interconnected . . . . . . . . .
Problem 5. Systems of Pipes in Different Planes
Interconnected . . . . . . . . .
7. Characteristics of Procedure
IV. METHOD OF BALANCING FLOWS . . . . . . . .
8. Statement of Method
9. Illustrative Problems
Problem 6. Simple Network-h varies as Q2 .
Problem 7. Multiple Inlets and Outlets-h varies
as Q2 . . . . . . . . . . . .
10. Remarks on Method. . . . . . . . . .
V. TYPES OF PROBLEMS ENCOUNTERED IN NETWORKS.
11. Typical Problems
VI. CONCLUDING REMARKS-OTHER APPLICATIONS OF METHODS
LIST OF FIGURES
NO. PAGE
1. Distribution of Flow in Single Circuit; Method of Balancing Heads . . . 10
2. Distribution of Flow in Simple Network; Method of Balancing Heads . . 13
3. Distribution of Flow in Network-Two Inlets; Method of Balancing Heads 14, 15
4. Distribution of Flow in Several Planes; Method of Balancing Heads . . 18, 19
5. Distribution of Flow in Several Planes; Method of Balancing Heads . 20,21
6. Distribution of Flow in Simple Network; Method of Balancing Flows . . 23
7. Distribution of Flow in Network-Several Inlets and Outlets; Method of
Balancing Flows . . . . . . . . . . . . . . . . . . 25
8. Types of Problems Encountered in Distributing Flow in Networks . . . 27
LIST OF TABLES
1. Problem 7-Successive Values of Inflow and Outflow . . . . . . . 26
SYNOPSIS
The problem of finding the distribution of flow in networks of
pipes occurs in the design of distribution systems for water. Similar
problems occur in connection with distribution systems for other
fluids such as steam or air and with electrical circuits. In general
the analysis of such systems by formal algebraic procedures is, if
the systems are complicated, very difficult. Models have been used
in studying this problem, but here, as elsewhere, there are some
objections to their use.
The methods here presented are methods of successive corrections.
The convergence is apparently sufficiently rapid in all cases to make
the methods useful in office practice. Perhaps the greatest value of
the methods is in training and assisting the judgment, but this, of
course, is the principal object of all analytical procedures.
Where the relation between flow and head (between current and
voltage) is linear, the corrections are exact. Where this relation
between flow and head is not linear, use is made of the relation
between increment of flow and increment of head, which relation is
linear for a given quantity of flow. If, however, the increments are
fairly large, this linear relation is somewhat in error. A sufficiently
exact solution is nevertheless obtained by successive correction. The
method thus involves an arithmetical application of the fundamental
principle of the differential calculus.
In the problems of flow discussed in detail in this bulletin it is
assumed that kinetic heads and head losses at junctions are small in
comparison with the lead losses due to friction, and that they may
therefore be neglected.
The problems under immediate discussion deal primarily with
systems carrying water and incidentally with those carrying electric
currents. Additional applications of the methods here presented are
suggested at the end of the paper.
Perhaps it should be added that this investigation, in a field of
study which is not the major interest of the author, is a by-product
of explorations in structural analysis.
The problems have been chosen to illustrate the method and not
because they represent any existing layouts.
ANALYSIS OF FLOW IN NETWORKS OF
CONDUITS OR CONDUCTORS
I. INTRODUCTION
1. Type of Problem.-As indicated later, various types of problems
occur in connection with flow in networks. In what is perhaps its
simplest form, the problem is usually as follows:
The quantity of fluid or of energy flowing into the system at one
point is known and the point of delivery is also known. The sizes
and lengths of the conductors or pipes in the system are given or
assumed, and also the law controlling the relation between quantity
of flow in the conductor and the loss of head or voltage in a given
length.
It is usually desired to determine the total loss of head or voltage
between inlet and outlet. If a single conductor connected these two
points, the loss of head for given flow could be computed directly
from the relation between flow and head loss. In a network, however,
this loss depends on the distribution of the flow in the system. If
such distribution is known, the drop of potential in each conductor
can be determined directly, and the total drop found as the sum of
the drops along any path connecting inlet and outlet, the total drop
being of course the same whatever path is chosen.
The difficulty arises in determining the distribution of flow in the
network. This is controlled by two sets of conditions, both simple
and obvious:
(a) The total flow reaching any junction equals the total flow
leaving it (continuity of flow)
(b) The total change in potential along any closed path is zero
(continuity of potential).
These sets of conditions, together with the relation between flow
and potential drop, lead to sets of equations in which either the flows
in the individual conductors or the potentials at the junction points
are taken as the unknowns.
If the flows are taken as the unknowns, the equations will be
those for continuity of potential; if the potentials are the unknowns,
the equations will be those for continuity of flow.* In either case,
*The two methods here presented represent general methods applicable to many engineering
problems. The physical conditions controlling engineering relations often consist of two groups of laws
which are quite independent of each other. In such cases either set of relations may be first expressed
in terms of the other, and then a formal or an approximate solution may be obtained to satisfy the second
condition. Compare the analysis of continuous frames by methods such as the theorem of three
moments, where the equations to be solved represent the geometrical relations, the unknowns having
been previously interrelated by statics, with the method of slope-deflection where the equations are
those of statics, the unknowns having been previously related from geometrical considerations.
ILLINOIS ENGINEERING EXPERIMENT STATION
the order of the equations will be that of the relation between flow
and loss of potential. If this relation is linear, the equations will be
linear. If, however, the relation is not linear, serious difficulties
arise in solving the equations.
In those cases where the relation between flow and change of head
is linear, the methods to be presented may be thought of as a book-
keeping procedure for solving linear equations; where the relation is
not linear, the method changes the problem into that of a succession
of linear relations by use of the fundamental principle of the dif-
ferential calculus.
2. Flow of Water in a Network of Pipes.-In any network of pipes
such as is shown in the problems discussed in the following pages,
it is known that in each closed circuit the sum of all changes in head
is zero, and that at each junction the quantity flowing into the
junction equals the quantity flowing away from the junction.
It is assumed further that we know the law determining the loss
of head in any length of pipe for a given flow. This law usually
takes the form
h = CVn
where h is the change in head accompanying flow in any length of
pipe, C is the loss in the pipe for unit velocity of flow, and V is the
velocity.
Since the quantity of water flowing in the pipe is A V, this relation
may be rewritten
h = rQn
where r is the loss of head in the pipe for unit quantity of flow. The
quantity r depends on the length and diameter of pipe and on its
roughness.
The problem is to find the amount of water flowing in each pipe.
When the distribution of flow is known, the losses of pressure
throughout the system are readily computed.*
It is important to note that, except as noted in the footnote,
only relative values of r are needed to determine distribution of flow.
II. METHODS OF ANALYSIS
3. Methods of Analysis Proposed.-Two methods of analysis are
proposed. In one of these the flows in the pipes or conductors of
*In ordinary cases (on the assumption h = rQ» with n the same for all pipes) the distribution
of total flow is independent of the quantity flowing. On certain assumptions as to the relation between
head and flow, this will not be true. The matter does not seem of immediate practical importance,
though it may be of scientific interest in some cases.
ANALYSIS OF FLOW IN NETWORKS
the network always satisfy the condition that the total flow into and
out of each junction is zero, and these flows are successively corrected
to satisfy the condition of zero total change of head around each
circuit. In the other method the total change of head around each
circuit always equals zero, and the flows in the pipes of the circuit
are successively adjusted so that the total flow into and out of each
junction finally approaches or becomes zero.
The former method is, for convenience of reference, here desig-
nated as the "Method of Balancing Heads," the latter as the
"Method of Balancing Flows." The method of balancing heads is
directly applicable where the quantities flowing at inlets and outlets
are known. The method of balancing flows is directly applicable
when the heads at inlets and outlets are known; in this case it will
probably be found more convenient than that of balancing heads.
In some problems it may be desirable to combine the two methods.
Both methods depend on the principle that the resistance to
change in flow in any pipe equals approximately nrQ(" - ) where
h = rQ".
III. METHOD OF BALANCING HEADS
4. Statement of Method.-The method of solution is as follows:
(a) Assume any distribution of flow.
(b) Compute in each pipe the loss of head h = rQn. With due
attention to sign (direction of potential drop), compute the total
head loss around each elementary closed circuit 1h = IrQ".
(c) Compute also in each such closed circuit the sum of the
quantities R = nrQ(n - 1) without reference to sign.
(d) Set up in each circuit a counterbalancing flow to balance the
head in that circuit (to make IrQ" = 0) equal to
SrQ" (with due attention to direction of flow)
ZnrQ(" 1) (without reference to direction of flow)
(e) Compute the revised flows and repeat the procedure.
Continue to any desired precision.
In applying the method it is recommended that successive com-
putations of the circuits be put on identical diagrams of the system.
In office practice such diagrams will usually be white prints. Write
in each elementary circuit the value IR, and outside the circuit write
first (above) the value 2h for flow in a clockwise direction around the
circuit and second (below) the value Zh for flow in a counterclock-
wise direction around the circuit. On the right of these figures put
an arrow pointing ) or ) to the larger figure. This arrow will show
10 ILLINOIS ENGINEERING EXPERIMENT STATION
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ANALYSIS OF FLOW IN NETWORKS
correctly the direction of counterflow in the circuit. This technique
is illustrated in the problems following.
5. Proof of Method.-If the distribution of flow assumed in the
first place were correct, the change of head around any single closed
circuit would be zero. This change of head is IrQ". Considering
for the present a single circuit, write for each pipe
Q = Qo0 + A
Then, rQn = r (Qo + A)" = r (Qn + nQ (-1) A + )
If A is small compared with Qo the remaining terms in the expansion
may be neglected.
Then, for
SrQ" = 0
SrQ' = -Ao2nrQ"n - 1)
STh yrQn
A = -- =- ---
IR InrQgn - 1)
Of course, if A is relatively large compared with Qo and n is greater
than unity, the approximation is not very good, but this is less
important than it might at first seem, because in any case we must
correct for the unbalanced head produced in one circuit by corrections
in the adjacent circuits, which in general requires a recomputation of
all circuits. The convergence is, for practical purposes, sufficiently
rapid.
6. Illustrative Problems.-
Problem 1.-Single Closed Circuit
(Figure 1)
This problem shows the elementary procedure for a single circuit
in each of three cases: (a) where h varies as Q (streamline flow, or
electrical resistance, E varies as I); (b) where h varies as Q1-5 merely
as an illustration of a fractional exponent; and (c) where h varies as
Q2, a common approximate value for water circuits.
In all of these cases it is required to distribute a flow of 100
between two pipes, one of which is four times as long as the other,
but which are otherwise alike. In each case, also, to show the con-
vergence, the first assumption is the worst possible, namely that the
total flow follows the longer path.
ILLINOIS ENGINEERING EXPERIMENT STATION
Of course each of these cases is readily solved directly. Thus, in
all cases let Qi be the flow in the shorter, and Q2 that in the longer
pipe. Then
(a) Q1 4 4
( - a QQ. = - 100 = 80
Q2 1 5
Q1 (4)% 2.52
(b) - Q, = - - 100 = 71.5
Q2 (1)% 3.52
Qi (4)M 2
(c) - Q1 = -. 100 = 66.7
Q2 (1) 3
The computations have in these cases been arranged in the order
explained already, which has been found very convenient, namely,
(1) Write the divisor ZnrQ(n - 1) within the circuit.
(2) Write on one side of the circuit first the sum ZrQ" for clock-
wise flow, and below this the sum SrQ" for counterclockwise flow.
If, then, the arrow indicating the direction of flow is written on the
right of these figures and pointing to the larger flow, it will correctly
indicate the direction of counterflow needed to balance the circuit.
In complicated problems observance of some such system is necessary.
Problem 2.-Simple Network-h Varies as Q2
(Figure 2)
This shows a very simple network with one inlet and one outlet.
It is here assumed that h varies as Q2. All pipes are assumed to be
alike. The purpose of the problem is to illustrate the method of
arrangement in a case slightly more difficult than that of a single
circuit.
Here, as before, a very bad first trial value was intentionally
chosen to illustrate the procedure. Of course, the exact solution is
at once known by inspection to be as shown, and almost any reasonable
trial converges rapidly.
The arrangement of computations is that previously recom-
mended. Note the relatively small change in 2,nQ( 1).
Procedure
Some distribution of flow (without excesses or deficiencies at the
junctions) is first assumed.
(1) Compute the unbalanced heads around each circuit. (In
this case h = Q2 in each pipe.) As previously noted, these are
written to one side of the circuit, first (above) the heads for clock-
wise flow within that circuit, next (below) those for counter clock-
ANALYSIS OF FLOW IN NETWORKS
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ILLINOIS ENGINEERING EXPERIMENT STATION
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ILLINOIS ENGINEERING EXPERIMENT STATION
wise flow. The difference is the unbalanced head. The arrow, )
or ), written to the right of these figures, points to the larger head
loss, and indicates the direction of counterflow.
(2) Compute the divisors 2,nrQ(n - 1). In this case nrQ(" - 1) = 2Q
in each pipe, since r = 1. These are written within the circuits.
(3) Revise the flows from these counterflows, and repeat the
process as often as required.
In the foregoing solution the method is intentionally applied
blindly, without any judgment at all. It is at once evident that all
circuits in the first trial are badly unbalanced, all flows in each circuit
being in one direction. Usually the circuits which are badly un-
balanced will at once be thrown more nearly into balance by guess.
Problem 3.-Complex Network-Two Inlets-h Varies as Q
(Figure 3)
This is a more complicated network, with two inlets and one
outlet. The solution involves no new principles. Here it is assumed
that h varies as Q. Of course, in this case the divisors for each circuit
are constant throughout.
No attempt has been made to compute fractional values. It is,
however, clearly possible to go to any reasonable degree of precision,
but with considerable increase in the computation required if results
correct to three figures are wanted.
The arrangement of computations follows that previously ex-
plained.
Problems 4 and 5.-Systems of Pipes in Different Planes Interconnected
(Figures 4 and 5)
In general, systems for distributing water in cities may, for pur-
poses of analysis, be considered as in a single plane. In other cases,
as, for example, in distributing steam or hot water to a heating system,
the distribution may take place in several planes, with interconnection
between the planar systems of distribution.
This type of problem presents no especially new features except
that successive distribution must be made in circuits closed by the
risers as well as in the circuits which lie in a plane. The pipes
chosen on each floor to close the circuits containing the risers are
selected arbitrarily. It will be noted that in such problems any pipe
may lie in only one circuit (an outside pipe in a floor) or in two
circuits, three circuits, or even in four circuits (two floor circuits and
two riser circuits). The total change in flow in the pipe is the sum
of the changes in all the circuits of which it is a member.
ANALYSIS OF FLOW IN NETWORKS
Problem 4 shows a rather impractical layout selected for simplicity
of illustration. The distribution is carried through only two steps to
show the procedure.
Problem 5 differs from Problem 4 only in having more risers.
Clearly the technique used in recording the flows is a matter of
individual choice. Some may prefer to use isometric diagrams
throughout the analysis.
It is believed that the diagrams are self-explanatory.
7. Characteristics of Procedure.-Certain characteristics of the
procedure will be noted. When the flow is adjusted in any circuit
the flow is increased in some pipes and decreased in others, so that
the quantity IR = InrQ(n - 1) is not very much changed, and need
not usually be recomputed for each change of flow.
Since the first adjustments are in a sense preliminary, it is useless
to attempt precision in making them.
The answer, when finally obtained, is inevitably correct, since it
satisfies the conditions that the quantities balance at each junction,
and that the heads balance around each circuit. Moreover, errors in
the procedure are not cumulative, and, if made, are ultimately
eliminated.
IV. METHOD OF BALANCING FLOWS
8. Statement of Method.-In the method of analysis by balancing
heads just presented, the flow at any junction is balanced throughout
the analysis, but the head around any circuit is balanced by successive
correction.
Another method is to keep the head balanced around any circuit
throughout the analysis, in which case the flow at the junctions is
balanced by successive correction.
We may, then, assume a series of heads throughout the system
and compute the flow in each pipe corresponding to the differences
of head. From these find the total flow to each junction except inlets
and outlets. Distribute this flow to the pipes connecting at the
junction in inverse proportion to their resistances. (R = nrQ(n - 1)).
This, of course, causes an excess (or deficiency) of flow at the next
junction, but by successive distribution the excess flow will ulti-
mately be squeezed out at the inlets and outlets of the system.
Note, in the first place, that if the flow is distributed as in the
ILLINOIS ENGINEERING EXPERIMENT STATION
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