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I LLINOIUNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN

PRODUCTION NOTEUniversity of Illinois at

Urbana-Champaign LibraryLarge-scale Digitization Project, 2007.

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James E. Statmeyer

UNIVERSITY OF ILLINOIS BULLETINIsSUED WEEKLY

Vol. XXXIV November 13, 1936 No. 22[Entered as second-class matter December 11, 1912, at the post office at Urbana, Illinois, underthe Act of August 24, 1912. Acceptance for mailng at the secial rate of postage providedfor in section 1103. Act of October 3, 1917, authorised July 31, 1918.]

ANALYSIS OF FLOW IN NETWORKS OFCONDUITS OR CONDUCTORS

BYHARDY CROSS

BULLETIN No. 286ENGINEERING EXPERIMENT STATION

PUBLIsHED Br TH UMNIVRSIT OF ILLINOIS, URBANA

Puca: Tranr-rnu COrra


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T HE Engineering Experiment Station was established by actof 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 theState.

The management of the Engineering Experiment Station is vestedin an Executive Staff composed of the Director and his Assistant, theHeads of the several Departments in the College of Engineering, andthe Professor of Industrial Chemistry. This Staff is responsible forthe establishment of general policies governing the work of the Station,including the approval of material for. publication. All members ofthe teaching staff of the College are encouraged to engage in scientificresearch, either directly or in co6peration with the Research Corpscomposed of full-time research assistants, research graduate assistants,and special investigators.

To render the results of its scientific investigations available tothe public, the Engineering Experiment Station publishes and dis-tributes a series of bulletins. Occasionally it publishes circulars oftimely. interest, presenting information of importance, compiled fromvarious sources which may not readily be accessible to the clienteleof the Station, and reprints of articles appearing in the technical presswritten by members of the staff.

The volume and number at the top of the front cover page: aremerely arbitrary numbers and refer tothe general publications of theUniversity. Either above the title or below the seal is given the num-ber of the Engineering Experiment Station bulletin, circular, or reprintwhich should be used in referring to these publications.

For copies of publications or for other information addressTHE ENGIN'EERING EXPERIMENT STATION,,

SUNIVERSITY OF ILLINOIS,URBANA, ILLINOIS


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UNIVERSITY OF ILLINOISENGINEERING EXPERIMENT STATION

BULLETIN No. 286 NOVEMBER, 1936


BY

HARDY CROSSPROFESSOR OF STRUCTURAL ENGINEERING

ENGINEERING EXPERIMENT STATIONPUBLISHED BY THE UNIVERSITY OF ILLINOIS, URBANA


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UNIVERSIIYOF ILLINOIS4000-7-36-10216 6 - 'NIOR


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CONTENTSI. INTRODUCTION . . . . . . . .1. Type of Problem. . . . . .2. Flow of Water in a Network of Pipes

II. METHODS OF ANALYSIS . . . .3. Methods of Analysis Proposed

PAGE778

* . . . 8. . . . 8

III. METHOD OF BALANCING HEADS . . . . . . . .4. Statement of Method5. 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 Q2Problem 3. Complex Network-Two Inlets-hvaries as QProblem 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 Method9. 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


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LIST OF FIGURESNO. PAGE1. Distribution of Flow in Single Circuit; Method of Balancing Heads . . . 102. Distribution of Flow in Simple Network; Method of Balancing Heads . . 133. Distribution of Flow in Network-Two Inlets; Method of Balancing Heads 14, 154. Distribution of Flow in Several Planes; Method of Balancing Heads . . 18 , 195. Distribution of Flow in Several Planes; Method of Balancing Heads . 20,216. Distribution of Flow in Simple Network; Method of Balancing Flows . . 237. Distribution of Flow in Network-Several Inlets and Outlets; Method of

Balancing Flows . . . . . . . . . . . . . . . . . 258. Types of Problems Encountered in Distributing Flow in Networks . . . 27

LIST OF TABLES1. Problem 7-Successive Values of Inflow and Outflow . . . . . . . 26


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SYNOPSISThe problem of finding the distribution of flow in networks of

pipes occurs in the design of distribution systems for water. Similarproblems occur in connection with distribution systems for otherfluids such as steam or air and with electrical circuits. In generalthe analysis of such systems by formal algebraic procedures is, ifthe systems are complicated, very difficult. Models have been usedin studying this problem, but here, as elsewhere, there are someobjections to their use.

The methods here presented are methods of successive corrections.The convergence is apparently sufficiently rapid in all cases to makethe methods useful in office practice. Perhaps the greatest value ofthe methods is in training and assisting the judgment, but this, ofcourse, is the principal object of all analytical procedures.

Where the relation between flow and head (between current andvoltage) is linear, the corrections are exact. Where this relationbetween flow and head is not linear, use is made of the relationbetween increment of flow and increment of head, which relation islinear for a given quantity of flow. If, however, the increments arefairly large, this linear relation is somewhat in error. A sufficientlyexact solution is nevertheless obtained by successive correction. Themethod thus involves an arithmetical application of the fundamentalprinciple of the differential calculus.

In the problems of flow discussed in detail in this bulletin it isassumed that kinetic heads and head losses at junctions are small incomparison with the lead losses due to friction, and that they maytherefore be neglected.The problems under immediate discussion deal primarily withsystems carrying water and incidentally with those carrying electriccurrents. Additional applications of the methods here presented aresuggested at the end of the paper.

Perhaps it should be added that this investigation, in a field ofstudy which is not the major interest of the author, is a by-productof explorations in structural analysis.

The problems have been chosen to illustrate the method and notbecause they represent any existing layouts.


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I. INTRODUCTION1. Type of Problem.-As indicated later, various types of problems

occur in connection with flow in networks. In what is perhaps itssimplest form, the problem is usually as follows:

The quantity of fluid or of energy flowing into the system at onepoint is known and the point of delivery is also known. The sizesand lengths of the conductors or pipes in the system are given orassumed, and also the law controlling the relation between quantityof flow in the conductor and the loss of head or voltage in a givenlength.

It is usually desired to determine the total loss of head or voltagebetween inlet and outlet. If a single conductor connected these twopoints, the loss of head for given flow could be computed directlyfrom the relation between flow and head loss. In a network, however,this loss depends on the distribution of the flow in the system. Ifsuch distribution is known, the drop of potential in each conductorcan be determined directly, and the total drop found as the sum ofthe drops along any path connecting inlet and outlet, the total dropbeing of course the same whatever path is chosen.

The difficulty arises in determining the distribution of flow in thenetwork. This is controlled by two sets of conditions, both simpleand obvious:

(a) The total flow reaching any junction equals the total flowleaving 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 flowand potential drop, lead to sets of equations in which either the flowsin the individual conductors or the potentials at the junction pointsare taken as the unknowns.

If the flows are taken as the unknowns, the equations will bethose 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 engineeringproblems. The physical conditions controlling engineering relations often consist of two groups of lawswhich are quite independent of each other. In such cases either set of relations may be first expressedin terms of the other, and then a formal or an approximate solution may be obtained to satisfy the secondcondition. Compare the analysis of continuous frames by methods such as the theorem of threemoments, where the equations to be solved represent the geometrical relations, the unknowns havingbeen previously interrelated by statics, with the method of slope-deflection where the equations arethose of statics, the unknowns having been previously related from geometrical considerations.


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ILLINOIS ENGINEERING EXPERIMENT STATION

the order of the equations will be that of the relation between flowand loss of potential. If this relation is linear, the equations will belinear. If, however, the relation is not linear, serious difficultiesarise in solving the equations.

In those cases where the relation between flow and change of headis linear, the methods to be presented may be thought of as a book-keeping procedure for solving linear equations; where the relation isnot linear, the method changes the problem into that of a successionof 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 pipessuch 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 headis zero, and that at each junction the quantity flowing into thejunction equals the quantity flowing away from the junction.

It is assumed further that we know the law determining the lossof head in any length of pipe for a given flow. This law usuallytakes the form

h = CVnwhere h is the change in head accompanying flow in any length ofpipe, C is the loss in the pipe for unit velocity of flow, and V is thevelocity.

Since the quantity of water flowing in the pipe is A V, this relationmay be rewritten

h = rQnwhere r is the loss of head in the pipe for unit quantity of flow. Thequantity r depends on the length and diameter of pipe and on itsroughness.

The problem is to find the amount of water flowing in each pipe.When the distribution of flow is known, the losses of pressurethroughout 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 ANALYSIS3. 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 distributionof total flow is independent of the quantity flowing. On certain assumptions as to the relation betweenhead 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.


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ANALYSIS OF FLOW IN NETWORKS

the network always satisfy the condition that the total flow into andout of each junction is zero, and these flows are successively correctedto satisfy the condition of zero total change of head around eachcircuit. In the other method the total change of head around eachcircuit always equals zero, and the flows in the pipes of the circuitare successively adjusted so that the total flow into and out of eachjunction 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 isdirectly applicable where the quantities flowing at inlets and outletsare known. The method of balancing flows is directly applicablewhen the heads at inlets and outlets are known; in this case it willprobably 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 tochange in flow in any pipe equals approximately nrQ(" - ) whereh = rQ".

III. METHOD OF BALANCING HEADS4. 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 totalhead loss around each elementary closed circuit 1h = IrQ".

(c) Compute also in each such closed circuit the sum of thequantities R = nrQ (n - 1) without reference to sign.

(d) Set up in each circuit a counterbalancing flow to balance thehead in that circuit (to make IrQ" = 0) equal toSrQ" (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. Writein each elementary circuit the value IR, and outside the circuit writefirst (above) the value 2h for flow in a clockwise direction around thecircuit and second (below) the value Zh for flow in a counterclock-wise direction around the circuit. On the right of these figures putan arrow pointing ) or ) to the larger figure. This arrow will show


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10 ILLINOIS ENGINEERING EXPERIMENT STATION

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correctly the direction of counterflow in the circuit. This techniqueis illustrated in the problems following.

5. Proof of Method.-If the distribution of flow assumed in thefirst place were correct, the change of head around any single closedcircuit would be zero. This change of head is IrQ". Consideringfor the present a single circuit, write for each pipe

Q = Qo + AThen, rQ n = r (Qo + A)" = r (Qn + nQ(-1) A + )If A is small compared with Qo the remaining terms in the expansionmay be neglected.

Then, for SrQ" = 0SrQ' = -Ao2nrQ"n - 1)

STh yrQnA = -- =- --IR InrQgn - 1)

Of course, if A is relatively large compared with Qo and n is greaterthan unity, the approximation is not very good, but this is lessimportant than it might at first seem, because in any case we mustcorrect for the unbalanced head produced in one circuit by correctionsin the adjacent circuits, which in general requires a recomputation ofall circuits. The convergence is, for practical purposes, sufficientlyrapid.

6. IllustrativeProblems.-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, orelectrical resistance, E varies as I); (b) where h varies as Q1-5 merelyas an illustration of a fractional exponent; and (c) where h varies asQ2 , a common approximate value for water circuits.In all of these cases it is required to distribute a flow of 100between 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 thetotal flow follows the longer path.


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Of course each of these cases is readily solved directly. Thus, inall cases let Qi be the flow in the shorter, and Q2 that in the longerpipe. Then

(a) Q1 4 4( - a QQ.= - 100 = 80Q2 1 5Q1 (4)% 2.52(b) - Q, = - - 100 = 71.5Q2 (1)% 3.52Qi (4)M 2(c) - Q 1= -. 100 = 66.7Q2 (1) 3

The computations have in these cases been arranged in the orderexplained 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 theright of these figures and pointing to the larger flow, it will correctlyindicate 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 Q 2(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 bealike. The purpose of the problem is to illustrate the method ofarrangement in a case slightly more difficult than that of a singlecircuit.Here, as before, a very bad first trial value was intentionallychosen to illustrate the procedure. Of course, the exact solution isat once known by inspection to be as shown, and almost any reasonabletrial converges rapidly.

The arrangement of computations is that previously recom-mended. Note the relatively small change in 2,nQ( 1).

ProcedureSome distribution of flow (without excesses or deficiencies at thejunctions) is first assumed.(1) Compute the unbalanced heads around each circuit. (In

this case h = Q2 in each pipe.) As previously noted, these arewritten to one side of the circuit, first (above) the heads for clock-wise flow within that circuit, next (below) those for counter clock-


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wise flow. The difference is the unbalanced head. The arrow, )or ), written to the right of these figures, points to the larger headloss, and indicates the direction of counterflow.(2) Compute the divisors 2,nrQ(n - 1). In this case nrQ(" - 1) = 2Qin each pipe, since r = 1. These are written within the circuits.

(3) Revise the flows from these counterflows, and repeat theprocess as often as required.

In the foregoing solution the method is intentionally appliedblindly, without any judgment at all. It is at once evident that allcircuits in the first trial are badly unbalanced, all flows in each circuitbeing 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 oneoutlet. The solution involves no new principles. Here it is assumedthat h varies as Q. Of course, in this case the divisors for each circuitare 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 resultscorrect 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 interconnectionbetween the planar systems of distribution.

This type of problem presents no especially new features exceptthat successive distribution must be made in circuits closed by therisers as well as in the circuits which lie in a plane. The pipeschosen on each floor to close the circuits containing the risers areselected arbitrarily. It will be noted that in such problems any pipemay lie in only one circuit (an outside pipe in a floor) or in twocircuits, three circuits, or even in four circuits (two floor circuits andtwo riser circuits). The total change in flow in the pipe is the sumof the changes in all the circuits of which it is a member.


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Problem 4 shows a rather impractical layout selected for simplicityof illustration. The distribution is carried through only two steps toshow 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 diagramsthroughout the analysis.

It is believed that the diagrams are self-explanatory.7. Characteristics of Procedure.-Certaincharacteristics of the

procedure will be noted. When the flow is adjusted in any circuitthe flow is increased in some pipes and decreased in others, so thatthe quantity IR = InrQ(n - 1) is not very much changed, and neednot usually be recomputed for each change of flow.

Since the first adjustments are in a sense preliminary, it is uselessto attempt precision in making them.

The answer, when finally obtained, is inevitably correct, since itsatisfies the conditions that the quantities balance at each junction,and that the heads balance around each circuit. Moreover, errors inthe procedure are not cumulative, and, if made, are ultimatelyeliminated.

IV. METHOD OF BALANCING FLOWS8. Statement of Method.-In the method of analysis by balancing

heads just presented, the flow at any junction is balanced throughoutthe analysis, but the head around any circuit is balanced by successivecorrection.

Another method is to keep the head balanced around any circuitthroughout the analysis, in which case the flow at the junctions isbalanced by successive correction.

We may, then, assume a series of heads throughout the systemand compute the flow in each pipe corresponding to the differencesof head. From these find the total flow to each junction except inletsand outlets. Distribute this flow to the pipes connecting at thejunction in inverse proportion to their resistances. (R = nrQ(n - 1)).This, of course, causes an excess (or deficiency) of flow at the nextjunction, 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


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foregoing the head difference between any inlet and outlet is approxi-mately unchanged, the change of head in one direction along onepart of any path connecting two openings being equal and oppositeto the change of head along another part. Of course, the valuesR = nrQC" - 1)are not exact when n is greater than unity, because thevalues of Q are not exact. Hence revision of the values of R isnecessary.

Note also that the total flow into and out of the system is ingeneral not balanced in the beginning, but that any excess of inflowwill untimately be squeezed out by successive distributions. Similarly,if there is a deficiency, we may imagine that it will inflow at the inletsand outlets.

9. Illustrative Problems.-Problem 6.-Simple Network-h Varies as Q2

(Figure 6)The problem analyzed in Fig. 6 is the same as that previously

analyzed by balancing heads in Fig. 2. The elevation at the inletis assumed to be 100. The outflow takes place at one point as shown.It is assumed that in each pipe-run of the system the loss of headequals the square of the quantity flowing. The procedure is as follows:

Procedure(1) Assume heads arbitrarily for inlet and outlet. In this case

assume inlet at El. 100 and outlet at El. 0. Guess at heads at otherjunctions as closely as possible. These values are shown in circlesnear the junctions.

(2) Compute flow (or relative flow) in each pipe for heads as-sumed. In this case it is assumed that h = rQ2, and therefore that

(3) Compute excess (+) or deficiency (-) of flow at each junc-tion except inlet and outlet. Write these in the squares shown foreach junction.

(4) Distribute these excesses or deficiencies to the pipes con-necting at each junction in inverse proportion to R = nrQ(n - 1). Inthis problem all values of r are assumed equal to unity, except inthe corner pipes which are twice as long, and hence have r = 2.1Also n = 2. Hence the flows are distributed in proportion to -,rQwhere.Q is the flow already computed.

(5) Carry over in the pipes connecting at each junction (exceptinlet and outlet) the flows distributed from adjacent junctions to


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find new values of excesses and deficiencies.(6) Distribute the new unbalanced flows and repeat to conver-

gence, thus determining the relative amount of the total inflowflowing in each pipe.Problem 7.-Multiple Inlets and Outlets

(Figure 7)In this problem it is assumed that the elevations are known at

several inlets-namely, El. 40 at inlet A, and El. 30 at inlet B-andat several outlets-namely, El. 10 at outlets B and D, and El. 0 atoutlets A and C. It is desired to determine the distribution of totalflow at inlets and at outlets, and, in general, the distribution of flowin the system. For simplicity of presentation it is assumed that theresistances are the same in all pipe-runs. It is also assumed thatQ2h = -- in any pipe.1000

First assume heads at intermediate junctions as shown by figuresin the circles at each intermediate junction.

Compute flows-or relative flows-in each pipe. Here the flowis computed as Q = V 1000h.

Compute excess or deficiency of flow at each junction, and writethis in the rectangle near that junction. Distribute these unbalanced

1flows in proportion to - at each junction. The relative valuesQfor distribution at each junction are written on the pipes at thatjunction.

Compute the excesses and deficiencies produced by the new flowsand distribute these.

It may be assumed that in a problem of this type what is reallywanted is the distribution of outflow and of inflow. Table 1 showsthese flows after successive distributions in this case. The con-vergence here is slow and not very satisfactory, partly because thefirst guess as to intermediate heads is not very good. Nevertheless,the computations indicate quickly the approximate distribution offlow at inlets and outlets.

Some circuits will be found to be unbalanced as regards head.These may now be adjusted, if desired, starting with the distributionfound for total flow.

10. Remarks on Method.-In such relatively complex problems asthat just shown, it seems clear that the method of distributing flowshas advantages. In general, however, it is thought to be less simple,obvious, and expeditious than that of balancing heads.


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