capacity of vertical drainage stack

7/28/2019 Capacity of Vertical Drainage Stack

1/58

" * "

N B S M O N O G R A P H 31

C a p a c i t i e s o f S t a c k s i n S a n i t a r y D r a in a g e S y s t e m s F o r B u i l d i n g s

U .S . D E P A R T M E N T O F C O M M E R C E


2/58

T H E N A T I O N A L B U R E A U O F S T A N D A R D SF unctions and A ctiv ities

Hie functions of the National Bureau of Standards are set forth In theAct of Congress, March 3, 1901,as amended by Congress in Public Law 619, 1950. These include the developmen t and maintenance of thenational standards of measurement and the provision of means and methods for mak ing measurements con*sistent with thesestandards;the determination of physical constants and properties of materials; the developmentof methods and instruments for testing materials, de vices, and structures; ad visory se rvices to governmentagencies on sc ientific and technical problems; invention and development of de vices to se rv e special needs of theGovernment; and the developmen t of standardpractices, codes , andspecifications. The work includes basicand ap plied research, development, engineering, instrumentation, testing, evaluation, calibration se rvices, andvarious consultation and information se rvices. Research projects are also performed for other governmentagencies when theworkrelates to and su pplements the basic program of the Bureau or whe n the Bureau's uniquecompetence is required* The sco pe of activitiesis suggested by the listingof divisions and sections on theinsideof the back cover.Publications

The results of the Bureau's work take theform of either actual equipment and devices orpublished papers.These papers appear either in the Bureau's ow n series of publications or in the journals of professional andscientific societies. TheBureau itself publishes three periodicals available from the Government Printing Office:The Journal of Research, pu blished in four separate sections , presents complete scientific andtechnical papers;the Technical News Bulletin presents summary and preliminaryreportson work in progress; andBas ic RadioPropagation Predictions provides data for determining the best frequencies to us e fo r radio co mm unicationsthroughout the world, There are also fiv e series of nonperiodical pu blications: M on ographs, Applied Mathematics Series, Handbooks, M iscellaneous Publications, andTechnical Notes.Information on the Bureau'spublicationsca n be found in NB8 Circular 460, Publications of the NationalBureau of Standards ($1.25) and its Supplement ($1.50), available fro m the Superintendent of Documents,Gov ernment Printing O ffice, Washington 25, D.C.


3/58


4/58

Contents

Page1 . Introduction_________________________________ 11.1. Purpose and scope of paper_____________ 11.2. General principles of the design of building-drainagesystems. _____________________ 11.3. Definitions._.________________ 21.4. Nomenclature_________________________ 22. Previous research on capacities of drainage andvent stacks.________________________________ 32.1. National Bureau of Standards. ___________ 32.2. Universities___________________________ 63. Test equipment and procedures.________________ 83.1. Tests on interference of flows at junction ofdrainage stack and horizontal branches-.... 8a. Simulated stacksanitary-tee and long-turn T-Y fittings._,________ 8b. Prototypal stacksanitary-tee fittings 83.2. Tests on airflow in stack vent and in vent tohorizontal branch_____________________ 93.3. Miscellaneous tests._____________________ 10a. Distribution of air and water in crosssection of 3-in. drainage stack. _____ 10b. Velocity distribution of water in crosssection of 3-in. drainage stack._____ 10c. Vertical distribution of pneumaticpressures within 3-in. drainage stack____ 114. Analysis of flow conditions in plumbing stacks_ 114.1. Interference of flows at junction of drainagestack and horizontal branches_________ 11a. Simulated stack_____..___-________-.. 14b. Prototypal stack__________________ 15

4.2. Terminal velocities and terminal lengths__ 16a. Derivation of equation for terminalvelocities. _______________________ 16b. Derivation of equation for terminallengths__________________ 184.3. Flow capacities of multistory drainagestacks______-____-_-_-.._-...._________ 184.4. Air flow in drainage systems.___________ 194.5. Flow capacities of vent stacks___________ 205. Test results._._________________________ 205.1. Flow capacities at junction of drainagestack and horizontal branches_________ 20a. Simulated stack____________________ 20b. Prototypal stack__________________ 225.2. Air-flow measurements.__________________ 235. 3 Miscellaneous measurements-_____________ 27a. Distribution of air and water in crosssection of drainage stack_________ 27b. Velocity distribution of water in crosssection of drainage stack______ 28

Pa5. Test resultsContinued5.3 Miscellaneous measurementsContinuedc. Pneumatic pressures within drainagestack___.______________6 . Application of results of investigation.__________6.1. Permissible simultaneous rates of flow atjunction of drainage stack and horizontalbranches. ____________________________6.2. Loads on drainage stacks______________6.3. Loads on vent stacks___.________________6.4. Comparison of loads given by plumbingco de s with computed values for drainageand vent stacks._______________ _______a. Loading tables for drainage stacks.___b. Loading tables for vent stacks________7. Conclusions_____-___-_______-_-______________7.1. Interference of flo ws at junction of drainagestack and horizontal branches_________7.2. Flow capacities of drainage stacks,________7. 3 . Air flow in drainage systems._____________7.4. Loading tables--_______-__-.____________7.5. Miscellaneous phenomena________________a. Distribution of air and water in crosssection of drainage stack._________b. Velocity distribution of water in crosssection of drainage stack_________c. Pneumatic pressures within drainagestack____________________7.6. Need for further research_______________

a. Design-flow problems______________b. Flow-capacity problems_____________(1 ) Flow of air in venting systems_____(2 ) Interference of flows at stack-branch junctions, __ ____________ (3 ) Velocity distribution of water, andair-water distribution in crosssection of drainage stack______(4 ) Detergent foam in plumbing sys-tems_________________________8. References.__________________________9 . Appendix___________________________________9.1 . Individual measurements relating to interference of flows at junction of simulatedstack and horizontal branches_________9.2. Computation of the absolute roughness, & .,for cast-iron soil pipe._________________

9.3 . Effect of drainage-stack height on permissible loads__-__-______-_ -___________


5/58

Capacities of Stacks in Sanitary Drainage Systems for BuildingsRobert S. Wyly and Herbert N. Eaton

Some of the important results obtained in investigations of capacities of plumbingstacks in test systems at the National Bureau of Standards and elsewhere are discussed.Data are shown from experiments on the flow of water and air in such systems, andanalyses of certain flow phenomena are given. Methods are shown for applying theresults of research in hydraulics and pneumatics to the preparation of loading tables (fordrainage and vent stacks) suitable for use in plumbing codes. The need for additionalresearch in further improvement of plumbing codes is discussed.

1. Introduction1.1. Purpose and Scope of Paper

The lack of an adequate knowledge of thehydraulics and pneumatics of building-drainagesystems is one of the most serious handicapsunder which the writers of plumbing codes anddesign handbooks must work. A building-drainag e system cannot be designd with maximumeconomy unless the minimum sizes of pipes thatcan be used satisfactorily in its various partscan be computed. This is particularly true of amultistory system or of a system serving a lowbuilding which covers a large area, since manysuch systems are large enough and complexenough to afford opportunities for appreciablesavings through the application of design methodsbased on research.Attempts have been made to determine byexperiment the capacities of various componentsof building-drainage systems. The complexityof the flow phenomena in such systems makesit necessary to simplify test conditions. Unfortunately, this limits the usefulness of the results.Hence, it is not surprising that loading tables insome plumbing codes have been based more onexperience and opinion than on rational analysisand experiment. It is certain that the factorsof safety used in preparing some loading tableshave been excessively large. This leads directlyto oversizing of pipes.The purposes of this paper are:1 . To summarize certain aspects of experimental studies of the hydraulics and pneumaticsof plumbing systems conducted at the NationalBureau of Standards and other laboratories since1921 , with particular emphasis on some of theresults obtained at the National Bureau ofStandards during an investigation of the capacitiesof stacks and horizontal branches;2. to correlate laboratory data on the flow ofwater and air obtained in this and other investigations;3. to illustrate the application of researchresults to the computation of loads for drainageand vent stacks; and

4. to compare some of the loadings obtainedfrom tables in currently used plumbing codes withloadings computed by methods outlined in thispaper.The most recent laboratory investigationdescribed in this paper comprised four parts:1 . A study of the flow conditions at the junctionof a drainage stack and its horizontal branches,using a specially designed flow simulator (simulated stack) with sanitary-tee stack fittings;2. a study similar to (1), except that long-turnT-Y stack fittings were used;3. a study similar to (1), except that a multistory test system (prototypal stack), rather thanthe simulated stack, w as used; and4. a study of miscellaneous items, including airflow in drain and vent pipes; pneumatic pressuresin drainage stacks; air content of the layer ofwater flowing on the wall of a stack, and velocitiesin this layer. The prototypal stack was used inthis part of the investigation.1.2. General Principles of the Design ofBuilding-Drainage Systems

The drainage system of a building requires twodistinct sets of pipes. The first is required totransport the waste water and the water-borneliquid and solid wastes from the plumbing fixturesto the street sewer, septic tank, or other means ofdisposal. The second set of pipes, the ventsystem, is required to reduce the intensity ofpneumatic disturbances caused by the intermittent use of the plumbing fixtures. The ventsystem, if properly designed, will prevent excessive depletion of the water seals of the fixturetraps. These water seals must be maintained toprevent sewer air from entering the building.In order that the drainage and vent systems mayperform their functions in a manner that meetsgenerally accepted standards of performance andthat will not create health hazards or nuisanceconditions, the following broad requirementsshould be satisfied:1 . The dimensions and arrangement of the


6/58

various components of the piping systems shouldbe such that normal peak loads of water and water-borne wastes discharged into the drainage systemmay be carried away by gravity without creatingexcessive hydrostatic pressures, and that pneumatic-pressure fluctuations in horizontal branchesand fixture drains shall not seriously deplete thewater seals in fixture traps.2. The design should be such that noise andvibration due to the flowing water are reduced to apractical minimum.3. The various sloping pipes in the drainagesystem should be of such diameters and slopesthat self-cleansing velocities will be attained.4. Adequate, but not excessive, allowanceshould be made for the effects of fouling and corrosion in the piping systems and for cases whereadditional loads may be imposed on the system atsome future date.1.3. Definitions

The following definitions of some of the termsused in this paperare given in order to avoid confusion or possible misinterpretation of statementsmade in the paper. Except for those that aremarked with an asterisk, the definitions have beentaken from the American Standard NationalPlumbing Code A40.8-1955 [I]. 1A branch interval is a length of soil or wastestack corresponding in general to a story height,but in no case less than 8 ft within which thehorizontal branches from one floor or story of abuilding are connected to the stack.The building (house) drain is that part of thelowest pipingof a drainage system which receivesthe discharge from soil, waste, and other drainagepipes inside the walls of the building and conveysit to the building (house) sewer beginning 3 ftoutside the building wall.The ^drainage stack is the vertical main of adrainage system and may be either a soil or awaste stack.A drainage system (drainage piping) includes allthe piping within public or private premises, whichconveys sewage, rain water, or other liquid wastesto a legal point of disposal, but does not includethe mains of a public sewer system or private orpublic sewage-treatment or disposal plant.A fixture unit is a quantity in terms of whichthe load-producing effects on the plumbing systemof different kinds of plumbing fixtures areexpressed on some arbitrarily chosen scale.

A horizontal branch is a drain pipe extendinglaterally from a soil or waste stack or buildingdrain, with or without vertical sections or branches,which receives the discharge from one or more fixture drains and conducts it to the soil or wastestack or to the building (house) drain.Themain vent (referred to as the ven t stack inthis paper) is the principal artery of the ventingsystem, to which vent branches may be connected.* Figures in br ackets indicate the literature references on page 4 0 .

The *piezometric head is the height to whichwater would rise in a vertical tube connected atits lower end to a pipe containing water and openat its upper end to the atmosphere.Plumbing fixtures are installed receptacles, devices, or appliances which are supplied with wateror which receive or discharge liquids or liquid-borne wastes, with or without discharge into thedrainage system to which they may be directly orindirectly connected.A soil pipe is any pipe which conveys the discharge of water closets or fixtures having similarfunctions, with or without the discharge fromother fixtures, to the building drain or buildingsewer.A stack is the vertical main of a system of soil,waste, or vent piping.A stack vent (sometimes called a waste vent orsoil vent) is the extension of a soil or waste stackabove the highest horizontal drain connecting tothestack.* Terminal length as applied to drainage stacksmeans the distance through which water discharged into the stack must fall before reachingterminal velocity.* Terminal velocity as applied to drainage stacksmeans the maximum velocity of fall attained bythe water in the pipe, whether or not the crosssection of the pipe is filled with water.A trap is a fitting or device so designed andconstructed as to provide, when properly vented, aliquid seal which will prevent the back passage ofairwithout materially affecting the flow of sewageor waste water throughit.The trap seal is the maximum vertical depth ofliquid that a trap will retain, measured betweenthe crown weir and the top of the dip of the trap.A vent stack is a vertical vent pipe installed primarily for the purpose of providing circulation ofair to and from any part of the drainage system.(In this paperit is to be understood that the developed length of vent stack may include somepiping in other than a vertical position.)A vent system is a pipe or pipes installed toprovide a flow of air to or from a drainage system orto provide a circulation of air within such systemto protect trap seals from siphonage and backpressure.A waste pipe is a pipe which conveys only liquidwaste, free of fecal matter.Awet vent is a vent which receives the dischargefromwastes other than water closets.

1.4. NomenclatureThe following list of letter symbols used in theanalyses and equations appearing in this paperwill be useful to the reader. Insofar as possiblethe symbols used are in agreement with ASAstandard YlO.2-1958, Letter Symbols for Hydraulics.General terms:

A=area of cross sectionD=diameter


7/58

force< 7 = accelerationof gravityJi=headL=lengthpressurevolume rate of flow (discharge rate)B=radiusp=mass density (mass per unit volume)mean velocityelevation above datumSubscripts:1=refers to drainage stack2=refers to horizontal brancha refers to airrefers to terminal conditionsrefers to ventw=refers towaterSpecial terms:head-loss coefficient for bend.head-loss coefficient for flow throughpipe fittings and transitions in crosssectionfunction of several variablesenergy loss per unit volume of waterdue to flow resistance/=Darcy-Weisbachfriction coefficient^centripetal force required to cause anelementary mass of water to movein a curved pathA , f t =headloss due to bendHydrodynamic-pressure head causedby deflectionof streamhead-loss term in Darcy-Weisbach pipe-flow equation resulting from frictionA I pneumatic-pressure head in drainagestack

& 2 =piezometric head in horizontal branchmeasured with reference to elevationof stack-branch junctionAg=height to which water will rise in avent connecting to a horizontalbranch, measured with reference toelevationof stack-branch junctionA 4 pneumatic pressure head in vent connecting to horizontal branch& s =Nikuradse sand-roughness magnitudeterminal length (length of fall requiredfor water in drainage stack to reachterminal velocity)X a dimensionless constant determinedbyfrictional resistanceAmmass of elementary volume of waterkinematic viscosityReynolds number, a dimensionless flowparameter^==hydrodynamic pressure caused by deflection of stream (corresponding tothe head0functional symbolratio of area of cross section of waterstream in a drainage stack to totalarea of cross sectionof the stackradius of curvature of path followed bystream when deflectedhydraulic radius, the ratio of cross se ctionof the stream in a drain pipe toits wetted perimeterthickness of layer of water flowing onwall of drainage stackT 7 f =mean terminal velocity of water flowingon wall of drainage stackdeflection coefficient applicable to headloss in pipe bend

2. Previous Research on Capacities of Drainage and Vent StacksAttempts have been made to obtain knowledgeof the flow and pressure conditions in stacks andthus to offer a rational basis for computing stackloadings. This has been difficult, for not enoughhas been known about the conditions in stacks topermit the establishment of awholly satisfactorycriterion for stack capacity. Hunter at the National Bureauof Standards, Dawson and Kalinskeat the State University of Iowa, and Babbitt atthe University of Illinois have done research onstack capacities.

2.1. National Bureau of StandardsSome of the first clear statements regarding thenature of flow in drainage stacks and a definitionof stack capacity were reported by Hunter in1923 [2]. From tests, he found that the characterof flow in a partially filled vertical pipe variedwith the extentto which thepipe was filled. Forlow rates of flow, the water was entirely on thewall of the stack; but as the flow rate was increased, the frictional resistance of the air causedthe formation of short slugs of water. Increased

air pressure generated by the slugs caused thewater to be thrown to the wall of the stack eitherimmediately or after falling for some distanceseparated into streamlets in the center of the pipe.Hunter observed that slug formation occurred ina 3-in. stack when the water-flow rate was increased to a value such that the flowing wateroccupied to of the cross section of the stack.A further increase in rate of flow resulted in closelyspaced slugs that did not break up readily.Hunter believed that intermittent slug formationis partially responsible for the rapid oscillationsofpressure which occurin plumbing systems.Hydrostatic head can develop in a drainagestack only when it is filled with water at somepoint. In the case of a stack which receiveswater at one elevation only, this condition wouldfirst occur at the elevation of water entrance,where the downward velocity is least.The vertical component ofthe entrance velocitydepends on the rateof flow by volume, the cross-sectional area of the inlet, and the angle of theentrance. This points to the capacity of the


8/58

fitting as a measure of the practical capacity ofthe stack. (Hunter was probably thinking hereof one- and two-story stacks.)Hunter defined as the rate offlow in gallons per minute at which the water justbegins to build up in the stack above the inletbranch of the fitting when no water is flowingdown the stack from a higher level.Determinations of fitting capacity were madefor different type fittings on both 2- and 3-in.stacks with the water introduced at one level.It was observed that therate of flow that causedthe water to stand above the fitting inlet wasgreatlyin excess of the rate atwhich the tendencyto build up first appeared.Some testswere made with inlets at two levelsapproximately 1 1 ft apart. Various rates of flowwere introduced into the stack through the inlets.Any backflow into an L-shaped tube set in a sideinlet of the lower fitting was taken to indicatethat the fittingcapacity had been exceeded. Thetendency was towardan increase in capacity withwater introduced at two levels instead of one.Hunter's tests indicated that stack capacity mayincrease with the number of inlets until the pointis reached when the stack is flowing full throughoutits length. From this line of reasoning, he co ncluded that the fitting capacities which he haddetermined for stacks with inlets at one levelonly would be less than the capacities of stackshaving inlets atmore than one level, and that suchcould be safely, but not necessarily economically, utilized as forall heights of stack.From his tests, Hunter decided that safe valuesfor stack capacity could be computed from theequation

(i)where is expressed in gallons per minute, andin inches. Hegives the values=22.5 for 45 Y inlets, and&=11.25 for sanitary-tee inlets.

On the basis of this reasoning and the experimental results with 2- and 3-in. stacks, he gavea table of stack capacities in BH13 [3 ] (s ee table 1) .The tests which form the basis of table 1 weremade with water introduced through double-branch fittings at one levelonly. Hunter reported

no quantitative results for the tests in whichwater was introduced into the stack at more thanone level.An illustration of a case in which the fittingcapacity has been exceeded is shown in figure 1 .This photograph was taken in the course of themost recent investigation of stack capacities atthe National Bureau of Standards. It shows a3-in. double sanitary tee with 3-in. side inletsmade of transparent plastic material with itsinside dimensions closely simulating those of the

TABLE 1.(H unter)

Diameter o fstack

234568

Practical carrying capacitySanitary-teefittings

b4 5M O O18 028 040 572 0

Y or Y- and-H-bendfittings

> > 9 0t> 20 03 6 056081 01,440 For water introduced at one elevation only, through do uble-branfittings.b Carrying capacities fo r the 2 - and 3-in. stacks were determined by expement. The values fo r larger diameters were computed fromeq (1 ) .

FIGURE 1 .

Total flow from branches is approximately 2 0 0 gpm. No flow down stafrom higher levels.

corresponding metal fitting. Flows of about 10gpm are entering the fitting from each horizontbranch, but there is no flo w down the stack froabove the branch level. It will be observed thathe water is standing up in the stack somewhabove the level of the branches. AccordingHunter's formula (eq (1), using the value 11.2for & ), the capacity of the fitting is approximatel100 gpm. Therefore, it might be expected thaa flow of 20 0 gpmwould overload the fitting.


9/58

With further reference to Hunter's work [3 ],consideration of the design criterion that nohydrostatic head may develop in a drainagesystem indicates that carrying capacity for agiven diameter will be less in a horizontal sectionof the system than in a vertical section. Thissuggests separate consideration of vertical andhorizontal sections.The fact that an appreciable head of water inany portion of a drainage system into whichbranches discharge tends to impair the efficiencyof the drainage indicates a limiting condition onwhich to base computations of permissible capacities. Hunter felt that although an occasionalhead of water in certain portions of a drainagesystem may not be harmful, the assumption of acondition under which no head can develop inany portion of the stack or house drain is certainlya safe procedure and apparently the only onethat will permit a general application of testresults.One of the most significant criteria for drainage-stack capacity suggested by Hunter is that whereterminal velocity exists, a stack should not beloaded to such an extent that more than to ofthe cross section of the stack is filled with water.Some such limit is required to prevent the occurrence of serious pneumatic disturbances associatedwith excessive rates of water flow.

It is necessary to have information on velocitiesin drainage stacks before computing rates of flowthat will produce a water-occupied cross sectionbearing a given ratio to the stack cross section.Hunter [2 , 4 ] made measurements of terminalvelocities in vertical pipes, on on e occasion withthe pipes flowing full and on another with themflowing partially full.

He measured the rate of fall of water columnsin vertical pipes up to 100 ft in height, using1-, 2-, and 3-in. galvanized steel pipe. Hisexperimental procedure w as as follows: Gageholes were tapped in the pipe at intervals of 10ft. A quick-opening valve was installed at thebottom of the pipe. With the valve closed, thepipe w as filled with water to the level of thefirst gage hole. The valve was then openedquickly, and the time required for the water toflow out of the pipe was observed. This processwas repeated, the pipe being filled to the heightof each gage hole successively. To obtain theterminal velocity for each diameter of pipe, heplotted the lengths of the falling column asordinates and the times of descent as abscissas.This gave a straight line in the region where thewater was falling at terminal velocity, and theslope of this line gave the terminal velocity.In order to establish limits within which themeasured terminal velocities should lie, he co mputed terminal velocities for smooth and forvery rough pipe over a range of diameters of from

1 to 8 in . These curves, together with theexperimentally determined velocities, are shownin figure 2. The three experimental points lie

I 1. 5 2 3 4 5 6 7 8 9 1010

FIGURE 2.

between the two curves, as they should, and, aswould be expected, they are closer to the curvefor smooth pipe than to the curve for very roughpipe. Because of this satisfactory agreement ofthe experimental values with the values givenby the smooth-pipe curve, Hunter concluded thatthe smooth-pipe curve given in figure 2 mightbe used to set the upper limit of terminal velocitiesfor flow out of stacks completely filled with water.Hunter [2 ] also measured velocities in 2- and3-in. cast-iron stacks flowing partly full. He useda 3-in. stack 45 ft in height, open top and bottom,and introduced the water through a 45 double-Yfitting at the top of the stack. The velocitieswere measuredwith a pitot tube at the bottom ofthe stack. Various heights of fall of from 5 to 45ft were used. From these tests, hefound the terminal velocity of flow in the 3-in. stack to be about16.8 fps, attained in a height of about 15 ft for aflow of 100 gpm; and 32.8 fps, attained in aheightof about 45 ft for a flow of 200 gpm.

The tests on a 2-in. stack showed that in a fallof about 20 ft for a flow of 45 gpm, a terminalvelocity of 18.5 fps w as attained; and for a flowof 90 gpm, a terminal velocity of 24 fp s wasattained.Hunter remarks that with flows of 90 gpm inthe 2-in. stack and 200 gpm in the 3-in. stack,slugs of water completely filling a short length ofthe pipe occasionally formed, and the maximumvelocity of these slugs approached the maximumvelocity for a completely filled stack.

Up to this point, the discussion of Hunter'swork has been confined to drainage stacks. Healso studied problems of venting. It seems appropriate to mention briefly a method which heproposed for sizing vent stacks [3]. He suggestedthat the formula(y-a) (*-6)=c (2 )


10/58

be used. In this equation, is the volume rateof water flow in gallons per minute divided by7.5 (Hunter at that time having conceived of the"fixture unit" as a volumerate of dischargeequalto 7.5 gpm); is the length of vent stack (mainvent) in feet; and a, b, and c are constants. Thecurve is asymptotic to the lines andBy the use of certain experimental data, Huntercomputed values of a, 6 , and These values aregiven in table 2 for vent stacks and drainagestacks of the same diameter.

rate of flow into the stack at any one floor levmust be limited. They suggested thatif sanitartee fittings are used, the rate of flow introducewithin any one branch interval should not exceeone-third of the maximum for the stack; while45 Y connections are used, the flow rate intrduced within any one branch interval should blimited to one-half of the maximum for the stacTable 3 gives the maximum carrying capacitiof drainage stacks suggested by Dawson anKalinske.

TABLE 2.(Hunter)

TABLE 3.(D awson and Kalinske)

Diameter ofboth drainage stackandvent stack

34568

Values of constantsa

6810121 G

b

3327201612

C

5,78011,40020 , 28031, 24081, 04 0

2 .2 . UniversitiesDawson and Kalinske at the State Universityof Iowa prepared two reports on an investigationof stack capacities [5 , 6]. Measurements were reported of volume rates of water and air flow,pneumatic pressures, and maximum water velocities in stacks 3, 4, and 6 in. in diam and approxi

mately 30 ft high. Each stack discharged intoa short horizontal drain through a 90 bend.Various rates of flow were introduced through ahorizontal branch near the top of the stack, andpressures and velocities were measured at severalpoints down the stack.From their tests, Dawson and Kalinske concluded that where the water attains its maximumvelocit3r , a drainage stack should not be allowedto flow more than about one-fourth full. Theyfelt thatthis criterion is a reasonable choice, sincethe velocity in the building drain and buildingsewer flowing full but not under pressure somedistance from the base of the stack is roughly ofthe order of one-fourth of the terminal velocityin the stack. They had observed in tests that astack flowing too full caused considerable noiseand vibration, as well as pneumatic disturbances.Dawson and Kalinske observed, as did Hunter,that too great a rate of discharge into a drainagestack at any onelevel caused the stackto fill up atthat point due to the small vertical component ofvelocity of the incoming stream. Because of thepneumatic and hydraulic disturbances which arecreated by this condition, they concluded that the

Diameter ofstack

34568

Carryingcapacity

901 8 035 056 01,200

The carrying capacities given in table 3 diffconsiderably from those in table 1 for some caseThis difference may be explained partly on thbasis that table 1 reallygives capacitiesindicated by a filling up of the stack with watnear the fitting, while table 3 gives capaciticomputedfor a given water-occupied cross sectiowhere terminal velocity exists in the stack somdistance below the fitting. Thus, the two tablare based on different phenomena.The measurements of terminal velocities madby Dawson and Kalinske with a pitot tube ashown in table 4. From the velocities givenTABL E 4.

(Dawson and Kali nske)Diameter bof stack

3333444446666666

Dischargerate(measured)

45901 3 51809 013 519 024 030 0

1 1 516 522 03 4545 056 067 5

Thicknessof sheet(computed)

0.136.209.281.34 6.141.189.252.311.382.162.199.221.281.320.37 9.446

Terminalvelocity(measured)

11.515.417.619 .516.819.020.421.222 .012.314.417.421.725 .026.527.5

* Water introduced at on e elevation.Nominal diameters. Thickness of sheet computed on basis of actdiameter of standard-weight steel pipe.


11/58

TABLE 5.(Hunter)

36

Diameter * >of stack

3322

Di schargerate(measured)

1 0 020 04 590

Thicknessof sheet(computed)

0.219.215.133.215

Terminalvelocity(measured)

16.834.218.524.0

Water introduced at on e ele vation.* > Nominal diameter. Thickness of sheet computed on basis of nominaldiameter. No measurements on actual diameters were reported by Hunterfor these tests.

the table, the cross-sectional areas, of thesheets of water were computed from the knownvalues of the flow rates. The thickness, of thesheet in each case was then computed from theequation(3)

The values thus obtained are also given in table 4.The same computations were made from Hunter'sdata onterminal velocities in stacks flowing partlyfull and the results are given in table 5.Dawson and Kalinske [ 5 ] derived the followingexpression for terminal velocity, which isexpressedhere in the notation of the present paper forsimplicity of comparison with eq (37) :(4)

After analysis of the factors which affect thefriction coefficient in eq (4), they concluded [ 6 ]that terminal velocities could be computed fromthe equation(5)

in which is expressed in feet per second, ingallons per minute, and Z ? i in inches. The dataon measured velocities shown in tables 4 and 5have been plotted in figure 3. The curve in thefigure has been computed from eq (5).The problem of computing sizes of vent stackswas studied by Dawson and Kalinske, along withthe problem of sizing drainage stacks [6] . Theydeveloped a table of vent-stack sizes based on testdata from the 3-, 4-, and 6-in. drainage stacksdescribed above and on the assumption that theair flow carried by the vent stack is proportionalto the product of the terminal water velocitytimesthe cross-sectional area of the drainage stack notfilled with water.Babbitt of the University of Illinois publishedthe results of stack tests [7] , Much of his workrelated to variationsof pressure inthe stacks withflow. He gives a discussion of "The useful capacity of 2-, 3-, and 4-in. soil stacks," in which he

28

24

20

12

/o \ 2/5' ' f t )

I______i_ 1________!_20 40 60 80 10 0 12 0

Q , (gpm)/D, , in.140

F IGURE 3.Dawson. . . . . . _ . . . . _ _ _ 3-in. stackV, __do..._______________._____________ 4-in. stackA,.....do. . . . . _ . . . 6- in. stack,Hunter.__________________________..__ 2-in. stack,_-___do__________________......_. 3-in. stack

discusses the capacity of a soil stack to receive thedischarge from plumbing fixtures. He states thata 4-in. soil stack will probably take all the waterthat would be delivered to it in a 5-story building,a 3-in. soil stack will probably take all the waterthat would be delivered to it in a 3-story residence,and a 2-in. pipe is unsuitable to be used as a so ilstack. He found from tests on a 4-in. and on a5-in. drainage stack [8 ] that thecapacities were 20 0and 50 0 gpm, respectively.Babbitt alsomade tests [7 ] to determine the rateat which one horizontal waste pipe of the samediameter as the soil stack can discharge waterinto the soil stack through a sanitary-tee fittingwithout backing up water in the system at thepoint of junction. He gives the following values:"2-in. soil stack, 25 gal per min for 7 seconds;3-in. soil stack, 50 gal per min for 7 seconds; and4-in. soil stack, 10 0 gal per min for 7 seconds."

Based on his test data and on empirical reasoning, Babbitt g ives a series of tables for sizing ventpipes for 3- and 4-in. drainage stacks for variousvolumerates of water flow. However, h e gives noventing tables for larger diameters of drainagestacks.58403261-


12/58

In the most recent National Bureau of Standards investigation of stack capacities, a simulatedstack w as first used which provided greaterflexibility of experimental conditions and bettercontrol of pertinent variables than could be provided by the u se of a prototype. Figure 4 sh owsthis equipment in some detail. To connect thestack and the horizontal branches, sanitary-teestack fittings were u sed first, followed by long-turnT-Y fittings.In this part of the investigation as well as inthose parts described under sections 3.2 and 3.3,water w as delivered to the test system from aconstant-level supply tank. The rates of discharge to the stack and the horizontal brancheswere measured by means of calibrated orificemeters.

DISTRIBUTION

MANOMETER CONNECTIONS

FIGURE 4.

In the simulated stack, the layer of water flowing down the stackwas produced at the level othe stack fitting by installing, in the stack jusabove the fitting through which the horizontabranches discharged, a second piece of pipesmaller in diameter than the stack. As can bseen in figure 4, the stack terminated at its uppeend in a chamber or drum supplied with watearound its circumference through six 1-in. diampipes. This construction caused water to discharge through the annular space between thstack and the insertedpipe, thus forming a movinblanket of water which simulated the layer thaunder service conditions is assumed to be flowindow n the stack from higher floors. The diameteof the inserted pipe determined the thickness othe layer, and thevolume rate at which water waintroduced into the drum determined the velocitof flow.Numerous tests were made on the simulatestack with one- and two-branch flow, for pneumatic pressures in the stack from 0.3 to 2.0 in . owater below atmospheric. Three diameters ohorizontal branches were used on the 3-in. stackIX , and 3 in. Data were obtained for waterlayer thicknesses of 0.16, 0.35, and 0.58 in. Thusthe tests provided nine different combinations odimensions for which pressures and rates of flowwere measured in the stack and horizontabranches.

Because conditions existing in the simulatedstack tests did not exactly correspond to conditions in a typical multistory drainage stack iservice, a part of the investigation was repeateusinga prototypal stack with six branch intervalsa building drain, and a system of ventilating pipesThe important elements of this system are showin fig u re 5.In making these tests, the system was arrangeso that the pneumatic pressure in the drainagstack was transmitted from the stack throughspecial connection to the horizontal branch atpoint 4 drain diameters from the stack, as showin figure 6 . Thus, the pneumatic pressure actinon the water in the horizontal branch was thsame as that in the stack, assuming equal aspiraing effect s at the tw o branch connections. Measurements with no water flowing in the brancindicated that the aspirating effects at oppositsides of the fitting did not differ bymore than 0,in. of water for rates of flow of up to 10 0 gpm ithe stack. Therefore, it will be assumed thathe head of water indicated in the sight glasattached to the horizontal branch showed directlthe di fference between the total pressure in th


13/58

VALVE

FIGURE 5.1. 3-in. ga lva n ized- ste el drai nag e stack .2. Horizontal branch es with co n tro lle d supplyof water.3. Sanitarv crossesat 8-ft intervals.4 . 4-in. cast -i ron soil-pip ebuildingdr ain , 50 ft long at aslop e of Hin . per ft .6. 4- in . long -sw eep bend.

PIPE CONNECTING HORIZONTALBRAN CHE S ON OPPOSITE SIDES OF STACK.

7

1 SIGHT GLA SS

horizontal branch and that in the stack. Thiseliminated the need for actually measuring thefluctuating pneumatic pressure in the stack. Itwas possible to obtain relatively steady readingsinmostof thetests bytheuseof thissystem.Water wasintroduced into the stack through adouble 3 - by 3 -in . sanitary-tee fitting (sanitarycro ss) 24 ft above the horizontal branch underobservation. All the water entering this fittingwasintroduced throughonesi de for flow s of up toabout 100gpm. Forflows ex cee din g 100 gpm, theexcess was introduced through the opposite si de.For each flow introduced at this point, severaldifferent rates of flow w er e introduced throughthe single branch under observation, this branchbeing connected to the stack through one si de ofa sanitary cross , as shown in fig ure 6. Branchdiameters of1 } and3 in. wereselected forobservation. Measurements w ere made of the rates offlow in the stack and in the branch, and of thehead of water in the sight glass. Water veloc ities in the stack at the level of the horizontalbranchwere not measured in these tests but werecomputedbythe methoddescribed in section4 .2 .of this paper.

Measurements of air fl ow in a 3 -in . stack ventand in a 2-in . vent to a 3 -i n . horizontalbranchwere made on the testsystem shown in fig ure 5.Figure 7 show s in detail the-piping arrangementat the horizontal branch to whichairwas deliveredthroughthe 2-i n . vent. Measurements w ere madeof air fl ow in the vents and of the pressure dropacross thelayerof waterflow in g onthe wall of thestack.

L*44PRIMARY ELEMENTOF AIR-FLOW -MEASUR ING DEVICE

TOP OFSTACK VENT

3" STAC K

3"HORIZONTAL BR ANCHFIGURE 6. FIG URE 7 .


14/58

Water was introduced 32 ft above the base ofthe stack (except in a few tests in which thisdistance was 48 ft) through one side of a 3- by3-in. sanitarycross, and the flow ofair was measured in the stack vent and in the vent to thehorizontal branchshown in figure7. Thisbranchwas at an elevation 24 ft above the base of thestack. Sanitary crosses wereinserted atintervalsof8 ft in this installation, as shown in figure 5.Before beginning these tests, all the vents wereclosed except the stack vent andthe vent connecting to the horizontal branch at the 24-ft level.The pneumatic pressurein thishorizontal branchcouldbe controlled at will by the degree of closureof a valve placed in the vent (see figure 7). Thestack vent wasfully open in some of the tests andcompletely closed in others. Each horizontalbranch, except the one through which the waterwas introduced, was closed. A device utilizingthe principle of the hot-wire anemometer was usedfor measuring the air velocity at the axis of thevent to the branch and at the axis of the stackvent. Because the primary elementof thisdevicewas placed at the axis of the pipes, a correctionwas applied to the measured values in order toobtain approximately the mean velocities insteadof the velocities at the axis. In making thesecorrections, mean velocity was assumed to be 0.8of the axial velocity, this value being in approximate agreement, for a value of the Darcy-Welshach friction coefficient /of 0.03, with anequation given by Rouse [9 ]. The volume ratesof air flow were computed in accordance with theformula,Insomeof thetests, the differencebetween thepneumatic pressure in thehorizontal branch andthat in the opposite sideof the3- by34n. sanitarycrosstowhich thehorizontal branchwasconnectedwas measured whenair wasflowing in thebranch.The inclined differential manometerused forthesemeasurements indicated approximatelythe difference between the pneumatic pressure within thebranchand that in the interiorof the stackat thelevel of the branch. A small correction wasactual!}' applied, obtained by noting thereadingon themanometer when water wasintroduced intothe stack whilepreventing the entrance of air intothebranch. Theneed for this correction probablywasthe result of a slight differencein the aspiratingeffect of the flow of water past the oppositeopenings of the sanitary cross. The correctionvariedfrom 0.06 in. to 0.09 in. of water, increasing withrateof waterflow inthe stack.

In order to investigate the distribution of airand waterin the cross section of a drainage stack,the test system shown in figure 5 was employed.A volume rateof water flow of 10 0 gpm wasintro-10

duced 48 ft above the base of the stack, andsampling measurements weremadejust abovethebranch at the 16-ft level, approximately 31.5 fbelow thepoint where waterwas introduced. Althevents onthetest system wereopen; hence theflow conditions should have approximated thoseto be expected in serviceforaflow rate of 100 gpmFigure 8 shows the principal parts of the speciaequipment usedfor sampling the flow in the stackAn impact tube of Ke-in . insidediameter, insertedinto the stack withitsopening facing directlyupward, was used to make a traverse of thediameterThe impingement of flow ontothetipofthe impactube forced air and water through the tube anddelivered it at the base of the transparentcylindeshown in thefigure. At the beginning of a testthe transparent cylinder (closed at the upper end)was first filled with water and the reservoir owater around the base of the cylinder was filledto the overflow-weir level. As the impact tubedelivered a mixture of air and water at the baseof the cylinder, the air rose to the top and displaced the water therein until the cylinder contained only air. The water displaced from thecylinder, plus thatdelivered the impact tubewas caught in a graduated measure. Thus, asthe quantity ofwater originally contained bytheinverted transparent cjdinder was known, and athe quantity of water flowing over the weir, andthe time required to displace a given volume owater in the cylinder could easily be measuredit was a simple matter to compute the averagerates at which air and water were delivered tothe cylinder.

00

FIG URE 8.1 . 3- in . drainage stack.2. Sampling tube.3. Ho se clamp.4. Circular openings.5. Petcock and con nection tovacuum sou rce.6. Overflow weirat same elevation as tipof sampling tube.7. Graduatedmeasuringve ssel.

Dataon the distributionof water velocity acrossa diameter of the 3-in. stack shown in figure 5


15/58

were obtained for water flows of 60, 80 , and 100gpm. These data were taken at the same pointat which the data describedunder (a ) above weretaken.A static-pressure tube wasinserted through thewall of the stack for thepurposeof obtaining thepneumatic pressure inside the stack, which wasindicated on an inclinedmanometer. The impacttube described under (a) above was connected toa single-leg manometer. The impact tube hadbeen calibrated by towing it through still waterat known velocities. By taking the differencebetween the readings obtained for the impacttube and those obtained for the static-pressuretube andreferring to the calibration curve, correspondingvelocities were indicated.

The ef fect of rate and vertical distribution ofwater flow on pneumaticpressureswithin adrain

age stack was studied experimentally by meansof the test systemshown in figure 5. In the firstseries of tests, water was introduced at one point32 ft above thebase of the stack. In the secondseries, water was introduced simultaneously attwo points, 32 ft and 16 ft above the base of thestack. In eachof these two series of tests, measurements of pneumaticpressureswithin the stackwere made by means of a small static-pressuretube insertedthroughthesideof the stacksuccessively at a number of points distributed vertically.Data were obtained both withthe venting systemfunctioning and with all vent pipes except thestack vent closed by valves. Water-flow rates of60 and 100 gpm were introduced at the higherpoint, andrates of 30 and 60 gpm were introducedat the lower point. In each case, the flow wasdeliveredthroughone side of a 3- by3-in. sanitarycross.

The flow from a horizontal branch into a multistory drainage stack will encounter a resistancewhen it meets the high-velocity blanket of waterwhich may simultaneouslyflow downward on thewall of the stack. This resistance creates abackpressure (head of water) in the branch as a resultof momentum changes in both thehorizontal andthe vertical streams. These changes are causedby the deflection of one stream by the other (seefig. 9). If the two flows become sufficiently large,the backpressure becomesexcessive and interfereswith the normal operationof the drainage system.Theproblem is to estimate the maximumrates of flow which can occur simultaneously in the stackand the horizontal branch without creating anexcessive head of water in the branch. Theanalysis of the problem can be based on either thelaw of conservation of momentum or the law of conservation of energy. Because of the geometryof long-turn T-Y stack fittings, the law of co nservation of energy is easier to apply in the case ofsuch fittings than is the momentum law. Actually, the momentum law was used in an earlierpaper [10 ] which reported the results of tests onthe simulated stack using sanitary-tee fittings.The analysis of the problem made in this paperbegins with an applicationof thelawof conservationof energy.

Figure 10 showsthe various terms whichwillbeused in the analysis. The piezometer shown inthis figure is for the purpose of the analysis only.FIGURE 9.


16/58

I STACKPIEZOMETRICHEAD ,

PNEUMATICPRES SURE =P 4VENT

PNEUMATICPRESSURE =P, i

ARBITRARY DATUMFIGURE 10 .

It show s the peizometric head (the height to whichthe water in the branch would rise through a tubeopen to the atmosphere at the upper end). Thishead may or may not be the same as the height towhich water rises in the branch or in a vent pipeconnected thereto when excessive rates of flowoccur simultaneously in the stack and horizontalbranch. The extent to which the piezometrichead and the actual water level inside the pipediffer depends on the pneumatic head in the vent/ & 4 , which may be either positive or negative.Pressures will be expressed in height of watercolumn through the relationship Theanalysis will begin by considering the change inenergy of the stream of water flowing in the branchas it passes from section b-b (mean elevation tosection a-a (mean elevation zj.The mean flu x of energy through the upstreamsection b-b per unit volume of water flowing inthe branch is

and the flux of energy through the downstreamsection a-a at the plane of impact with the flowingwater in the stack is

(|fwhere is the mean velocity and the meanpressure head at the section of impact of the twostreams.From continuity considerations, it appearsreasonable to assume that is approximatelyequal to and it is convenient to assume thatthe effective mean pressure head over the

impact section can be represented by the sum othe pneumatic pressure head, in the stack anddynamic pressure head, related to the curvlinear flow produced by the deflection of the stacstream by the branch stream. Thus, it is assumethat and Under these assumptions, the flux of energy throughsection a-can be expressed as

The difference between the unit flux of energy asections b-b and a-a represents the amount oenergy lost by a unit volume of water in passinfrom theupstream section to the downstream section. This energy loss is caused by friction in thhorizontal branch and stack fitting and by thchange in direction of flowproduced by the fittingand may be expressed as(6

The energy loss per unitvolume of water flowinthrough a straight pipe due to friction can bexpressed aswhere A/ is the head loss computed from the DarcyWeisbach formula for pipe flow.The energy loss per unit volume due to thepresence of a bend in a pipe can be expressed as

where is the head loss due to the bend. Inlong pipe having a bend in an intermediate location, the bend loss is the sum of two componentson e of which is due solely to the deflection of thestream filaments by the bend (the deflection component) and the other of which is due to the velocity disturbance caused by flow around the bendand which extends a considerable distance downstream of the bend (the tangential component)In the case under consideration the downstreamsection is missing, for all practical purposesTherefore, it is reasonable to assume here thaonly the deflection component f of the bend-loscoefficient is effective. This assumption makes ipossible to express the head loss between the upstream and downstream sections caused by pipefriction and by deflection of the stream filamentsby the bend as

(7)

Therefore, the energy loss per unit volume of waterflowing between sections b-b and a-a due to thetwocauses may be expressed as(8)2 < 7


17/58

Substituting in eq (6) , the expression(see fig. 10), substituting the right member of eq (8 ) for and dividing by thefollowing equation is obtained:(9 ) 2

All the quantities in eq (9 ) can be measured orestimated approximately except the dynamic head,The interaction between the layer of waterflowing in the stack and the flow from the branchis too complicated to analyze, except in a verygeneral and approximate way. In making thisanalysis, it is assumed that the flow lines in thelayer of water flowing down the stack are changedfromstraight lines to arcs of circlesof radius bythe stream from the branch (see fig. 11). Obviously, this is a simplifying assumption that is notentirely correct, yet it does result in an equationwhich agrees fairly w ell with data obtained byexperiment over a wide range of conditions.Under this assumption, summing up the centripetal forceswhich the flow from the branch exerts

on all the elementary volumes in the water layerin the stack subject to deflection (figs, llaandlib) resultsin the expression:(10)

where Am is the mass of the elementary volumeTAsA&. Then, if an average value of for allelementary volumes is assumed,'

It follows thenthat(12)

and, substituting this result in eq (9) , the followingequation is obtained:(13)

Dividing by and rearranging terms givesthe equation(f

All of the quantities in eq (14), except the radiusof curvature of the flow lines, aresubject to deter- This equation is identical to eq (27) of th e earlier paper [ 1 0 ] except for theinclusion of the bend-loss coefficient in eq (9 ) above. In th e earlier paper,which reported results of an experiment using sanitary-tee fittings only, itwas assumed that the bend loss du e to the geometry of the fitting wa snegligible.

FI GURE 11 .a an d b. Deflection of stream in stack by that in branch.c an d d. Deflection of stream in branch by that in stack.

mination, by measurement or otherwise. Thequantity will now be considered in more detail.Other things being equal, it will be assumedthat is directly proportional to the momentum,of the layer of water flowing down thestack, and that is inverselyproportional to themomentum, of the water in the branch.Thevalidity of these and other assumptions madein evaluating will be determined by thesuccessobtained in fittingthe final equation to the experimental data. Based on theabove considerations,

But must be placed equal to a length timesthis function if the equation is to be dimensionallycorrect, and the dimension that seems to be mostclosely involved with is the diameter, Z ? 2 , of thebranch. Hence,

where is the cross-sectional area of the waterflowing down the stack at the level of the branch,and is the cross-sectional area of the streamin the branch for the branch flowing full. Inorder to simplify the analysis, the area will bereplaced by its approximate equivalentThe latter expression will givevalues of area whichare accurate within 10 percent for stacks flowingnot over three-tenths full.Hence,

i ( F o q(16)13


18/58

the constant 4 being absorbed in the coefficientGoing one step further, it will be assumedthat the function is a simple power function.This was foundby logarithmic plotting of experimental data to be approximately true for theratio and will be assumed to be true for thetwo other ratios. Thus,o r

(17)

The value of given by eq (17) will now besubstitutedin eq (14) , writing for simplicity:

Rearrangement and simplification give theexpression:

(18)in which the coefficient is a constant.There are two quantities, and & , tobe determined from the experimental data. From thispoint on, the analysiswill be carried out separatelyfor the simulated stack and the prototypal stack.

In the caseof thesimulated stack, the construction of the test equipment was such that waterflowed down the stack in a laj^er which had, as itapproached the level of the horizontal branchunder consideration, a definite known thicknessand a definite known mean velocity of flow T 7 i(see fig. 4).In the analysis applying to the case in whichsanitary-tee fittings were used, the coefficient fappearing in eqs (9 ) and (1 8) was assumed to bezero. In other words, the resistance due to thedownward deflection of thehorizontal stream produced by the slightly curved passage through thefitting was neglected.In connection with the analysis applying tolong-turn T-Y fittings, a review of some of thepublished data on head losses in pipe bends wasmade. Measurements reported by Beij [11] on90 bends between upstream and downstreampipes showed that the average deflection losses inhis experiment were only about 37 percent asgreat as the average bend losses, the latter ofwhich included both deflection and tangentiallosses. If a roughness corresponding to0.00015 (as given by Rouse [9 ] for wrought-iron

14

or steel pipe) is assumed, a bend coefficient o0.27 for a radius of curvature of one pipe diameteis estimated for a 90 bend on thebasis of Beijresults. The radii of curvature of the bends ithe long-turn T-Y fittings used in the experimenreported herein were of the order of one pipdiameter. Various investigators have suggestespecific bend-loss coefficients for use with 45 an90 bends. The values suggested for 45 bendvary from 47to 75 percent of the values suggestefor 90 bends. Based on a bend-loss coefficienof 0.27 for a 9 0 bend as indicated above, ondeflection loss of 37 percent of the total loss duto a bend in a long pipe line, and on a ratio o45 bend loss to 90 bend loss of 0.47, a value of of 0.047 is computed for the case under consideration herein.

The term 7^- in eq (18) was held constant at/2value of 4.0 in the tests. Therefore, this valuwill be used in what follows.If values of 0.03, 0.00, and 4.0 for /, f, anare inserted in eq (18) as it applies to the casin which a sanitary-tee fitting is used, the equatiomay beplaced in the form

F=8.33-(7" (19In a similar fashion, if values of 0.03, 0.047and 4.0 for/, f, and are inserted in eq (18as itapplies to the case in which a long-turn T-Yfitting is used, the equation may be placed in thform

(20Further steps in the analysis will be simplifieby combining several of the terms appearing ieqs (19) and (20) as follows:

/n\ a \ /v\ ac"(f)(5;) (>='If 5 in eq (21) issubstituted in eqs (19 ) and (20the following equations are obtained

_ 8 .3 3 rQ _ -

v ! I ~*2) \^ /

for the sanitary-tee fittings, and 6.00-F

(2 2

(23for the long-turn T-Y fittings.The value of the exponent can be determinefrom experimental data by logarithmic plottinof values of computed from eqs (22) and (23against the variable -^- The slope of the line


19/58

plottedinthis way gives the approximate value ofThe value of the coefficient may then bedetermined by inserting in eq (21) the values of 5and together with the correspondingvalues of and and solving for These (determinations are given later in section 5.1.a, for both thesanitary-tee fittings and the long-turn T-Yfittings.

The analysis of flow interference at a branchconnection on a prototypal stackis made in sucha way that the dimension is not involved. It isassumed that the dynamic head is caused bythe deflection of the horizontal stream by thevertical stream, instead of conversely as in thecase of the simulated stack. It is assumed thatthe horizontal stream is deflected on an arc ofunknown radius of curvature, It is realizedthatpart of the deflection of the horizontal streamis caused by gravity. However, since this effectshould be small in comparison with the deflectioncaused by the interference of the two streamswhen there is appreciable flow in the stack, theeffect of gravity on the curvature of the streamlines w ill be neglected. The centripetal force,required to cause an elementary mass A mmoving at a velocity to follow a curved pathof radius of curvature will beT 7 2 (2 4)

If the elementary volume is taken as a cylinderof length A L cut perpendicular to the axis of thehorizontal branch, its mass A m will be

Therefore,

It is assumed that the average pressurerequiredto cause the curvature of the stream lines is equalto the applied force divided by one-half thecircumferential area of the elementary cylinder, 3 or

Thus,d=-pJDz or, dividing by

(25)3 This particula r assum ption is not essential to the solution of theproblem.The end result would be the same if it we re assumed that the pressure is equa lto the force divided by the area of the plane which is limited by the boundariesof the elementary cylinder an d which is bisected by its longitudinal axis.This wo uld yield the expression The disappearance of th e con-2stant - n thelatter case would be taken in to account in thevalues deter-T

It appears that, in general, will increasewhen the momentum per unit volume of waterin the horizontal stream increases and will decrease when the momentum per unit volume ofwater in the vertical stream increases. Themomentum of the water in the vertical streamwhich passes through a unit area inunittime is afunction of

and the momentum of the water in the horizontalstream which passes through a unit area in unittime is a function of

Hence,

Since must have the dimension of a length,and since the length which appears to be mostclosely associated with is the expression(2 6)

w ill be set up. Substitution in eq (25) of theexpression for given in eq (26) yields theequation(27)

If eq (27) is combinedwith eq (9), the followingequation is obtained relatingthe various heads

(28)If the function 0 is a simple power function,if /=0.03, if f=0, and if Z,/Z?,=4.0 (this valueof applying to the test equipment used),the following equation is obtained, after somerearrangement

r=8.33 ran2 9 )Now, if and are introduced in the sameway that they were in the development of eqs(19) and (20), eq (29) can be expressed as

r o oo_nr /^* - \~y (30)Th e end result of this analysis would be the same if it were assumed atthis point that the radius of cu rvature is a function o f the relative fluxof energy in the horizontal and vertical streams.


20/58

I feq (30) can be written as r=8.3367, or

8.33-F

(3 1 )

(32)

The value of the exponent can be determinedfrom experimental data by logarithmic plottingof values of computed from eq (3 2) against thevariable The slope o f the lines plotted in thisway indicates the approximate value o f a. Thevalue of the coefficient may thenbe determinedby inserting in eq (31) the values o f and a,together with the corresponding values of andand solving for These determinations aregiven later in section 5.1.b.

The following derivation o f the equation forterminal velocities is an approximate one, inwhich the water is treated as if it were a rigidbody sliding d o w n the stack instead o f a fluidlayer with a radial velocity gradient. A morenearly correct solution would require a consideration o f the turbulent boundary layer whichdevelops in the pipe. However, because of thedifficult} 7 of obtaining an exact solution, an approximate solution is given in this paper.The annular layer of water is treated as if itwere a rigid body moving down a plane verticalwall, acted o n only by the forces of gravity andwall friction. The effect o f the pneumatic-pressuregradient within the air core should be consideredin a rigorous solution o f the problem; but, sinc ein a properly vented stack the pneumatic-pressuregradient should be relatively small, this effectwill be neglected in what follows. It is assumedthat the water starts with an initial velocity F downward. The velocity F is less than the terminal velocity, which it will attain ultimatelyif the length o f fall is great enough. The thicknesso f the layer decreases with (the distance measured from the point of water entrance downwardto the point under consideration), becomingconstant when attains the value o f theterminal length.Since, under the assumptions made above, theaccelerating fo rc e is equal to the gravitationalforce less the frictional resistance,

A m -7T= (33)where T O is the wall shear per unit area, and theelementary volume o f water is assumed to beunits wide. But

80, (34

T O will now be defined by the relation,T O= ( 3 5

will be replaced by its value, FA/. Equation(33) then becomes

o r(36

The expression for the terminal velocity,will now be obtained by setting equal tozero. This yields the equation(37

It can be shown that X is equal to //4, where /the dimensionless friction coefficien t in the DarcyWeisbach formula for pipe fl ow. It is a functiono f the Reynolds number, approximately, and o f the roughness ratio, orIt will be necessaryto modify eq (37) to eliminate the friction co efficient X . To d o this, thfo l lowing equation given by Keulegan [12] foflow in open channels, based o n the Manninformula, is used:(38

where is the mean velocity in the cross sectioand is the shear velocity defined by threlation,^*=VWP^ (39

The Fin eq (38) is the same as the Fin the otheequations. T O is eliminated between eq s (3 8) an(39). If is replaced by as can be donapproximately, there results the equationX=0.0303 I ^ (40

Substituting this value o f X in eq (37), therresults

being written for to indicate that the value ois that corresponding to the terminal velocity.16


21/58

One further change will be made in the expression for terminal velocity by eliminating thethickness of the sheet of water by means ofthe relation between the thickness of the sheet,the cross-sectional area of the sheet, and the rateof flow. This relation is given by the approximateexpression (42)or by the more accurate expression

(43)Equation (42) should be sufficiently accurate forthe relatively small values of which are produced in drainage stacks flowing partly full. Ifthe approximate expression is used, there results

03x1/10 /0A20 (g)The reason that the Reynolds number is nota factor in eq (44) is that the relationship (e q

(38)) used to eliminate X is based on the Manningformula which is most accurate for conditionsunder which X is independent of the Reynoldsnumber. discussion by Rouse [9 ] indicatesthat the Manning formula is most dependable forintermediate values of relative roughness andleast dependable for low values of the Reynoldsnumber. Computations based on a cast-ironstack flowing with a water-occupied cross sectionof at least one-fourth of the cross section of thestack, and with a velocity in accordance with eq(44) indicate that X is substantially independentof the Reynolds number for stacks of 4 in. in diamor larger, and that the error in the use of theManning formula over the range of relativeroughness involved ranges from approximately7 percent for a 4-in. stack to approximately 3percent for a 12-in. stack. The low order ofmagnitude of this error indicates that, in viewof the proposed application of eq (44), the useof the Manning formula in its development isreasonable.An appropriate value of has been determinedfrom data on friction losses for flow through newcast-iron soil pipe tested by Hunter [13] . Thisyields the approximate value, 0.00083 ft.(See appendix, section 9.2., for the derivation ofthis value.) Rouse [9 ] gives the value, A:,=0.00085ft, for new cast-iron pipe.Equation (44) can be expressed in convenientform for computation by substituting thenumerical values of (0.00083 ft) and (32.2ft/sec/sec). The following equation (for cast-ironsoil pipe) is obtained

(45)where is expressed in feet per second,cubic feet per second, and Z > i is in feet.

If is expressed in gallons per minute and Z ? _in inches, eq (45) becomesV=3.0 (46)

Equation (46) is plotted in figure 1 2, togetherwith the experimental data obtained by Hunterand by Dawson and Kalinske. An average curvehaving the equation

(47)

is drawn through the experimental points.As would be expected, the experimental pointsscatter considerably, for the terminal velocity isvery difficult to measure. All the measurementslie above the curve computed from resistancemeasurements on cast-iron soil pipe. One possibleexplanation for this effect relates to the velocitydistribution in the layer of water. A steep velocitygradient exists in the layer, the velocity increasingfrom zero at the wall to a maximum at the innersurface of the sheet. The experimenters made

20 40 60 80 100Q! (gpm)/D, ,in. 12 0 140

based on flow-resistance measure

FIGURE 1 2.1. Average experimental curve,2. Computed curve,

ments in cast-iron soil pipe.9 , Dawson ____ _ _ __ _ _ _ _ _ ___ _ _ _ ___ __ _ _ _ _ _ ___ 3-in. stack., __o____ _ _ _ _ _ _ _______________ _ _ ___ 4-in. stack.A, __o__________-_,________ _ _ _ _ __ _ _ _______ 6-in. stack.8 , Hunter________.________._.__.________________-__ 2-in. stack., __o --------__--___________________ _ _ __ __ 3-in. stack.


22/58

their observations with pitot tubes with dynamicopenings pointing vertically upward through thevertical layer. It was impossible to get thepitot tube very close to the wall where the lowestvelocities existed, andhence, the velocities in thisregion werenot taken into account in the experimental measurements reported. An indirect indication that the velocities reported Dawsonand Kalinske may be too high can be obtained byinspection of their data [5], They computedvalues of Manning's of the order of 0.006 fromtheirexperimental measurements. This compareswith values of 0.012 to 0.015 for uncoated cast-iron pipe, and of 0.013 to 0.017 for galvanizedwrought-iron pipe given by Horton [14]. Allknown measurements on gravity flow in slopingpipes of various materials have yielded values ofsubstantially greater than 0.006. The surprisingly low value of indicated by the data ofDawson and Kalinskecould have been the resultof using velocity values that were too large incomputing If thereported velocities are substantially greater than the true velocities, asindicated by the above discussion, the curvecomputed from eq (46) may be more nearly correctthanare thedirect measurements.

On the other hand, it is known that a negativepneumatic-pressure gradient exists for a limiteddistance below a point of water entrance. Thistends to accelerate the water, and, if the acceleration were to be sufficiently great, velocities mightexceed those indicated eq (46). However, thedata of Dawson and Kalinske [5 ] on watervelocities as a function of distance below thepoint of water entrance indicate that the existenceof a negative pressuregradient in the upper partof the stack was not a significant factor in connection with terminal velocities. Since eq (46)is intended for application to well-vented stackswhere the pneumatic gradient is small in relationtothat which existed in the experiments reportedby Dawson and Kalinske, it seems reasonable toneglect the possible effect of the negative pneumatic gradient near points of water entrance onterminalvelocities.

The equation for terminal lengths in layer flowdown the walls of vertical stacks is derived bystarting with eq (36), making the necessary transformations, and integrating the result to obtainthe expression for the terminal length. It is noted first that

(48)If this substitution is made in eq (36) and theequation is solvedfor there results

, /1\ /wfryxA V\ 2 0 & / (49)

If it is assumed that X is not a function of aneq (37) is made use of,. ( F , ) 2 / ( F / T O ( W v ) \

I - __/TTTTq f I/ Ve d e ,( 5 0

whereEquation (5 0) can be integrated directly, witthe result:

6 3 g

-tan- 2 (51

If the upper limit is substitutedthere is obtained an infinite result, as the denomnatorofthefirst termin thebrackets becomes zeroThis difficulty is avoided by assuming that,whethe velocity has reached a value equal to 0.9times the true terminal velocity, the terminalength has been attained for all practical purposesMaking thissubstitutionfor in eq (51), aninserting the limits, there is obtained

1 ^

or

29,700-tan- 1 2.98

i=0.052 (52Equation (5 2) is useful for computing the distance through which water on the wall of a drainage stack must fall before it attains a velocitapproximately equal to the terminal velocityFigure 13 gives the terminal length as a functioofterminal velocity.Ifeq (52) iscompared with the equation of frefall under the influence ofgravity, itis found thathe layer of water in the pipe must fall approxmately three timesas far to attain a given velocitoffall as a body falling freely under theinfluencof gravity. Figure 14 showsterminal length asfunction of stack diameter, for two degrees ofullness. These curves were computed from eq(46), and (53). They indicate that inmo

practical cases the falling water will approacterminal velocity in a distance of from one ttwo stories.

As indicated in section 2, previous research hashown thenecessity for criteria for limiting thdischarge of water into drainage stacks. Thcriterion developed herein is based on 1 i T n i t . f t t . i Q18


23/58

050 15 20 25 30TERMINAL VELOCITY , f pS

FIGUEE 13.

345STACK DIAMETER , in.

FI GURE 1 4.

of the water-occupied cross section to a specifiedfraction, of the cross section of the stack whereterminal velocity exists, as suggested by earlierinvestigators.Plow capacity can be expressed in terms of thestack diameter and the water cross section. Thisexpression,

ft-27.8(rs ) 5 '3 (A) 8 / 3 (53)is derived by equating the fundamental expressionfor velocity, in which

AA)2 ,and the expression for terminal velocity given byeq (46). In eq (53), is the volume rate ofwater flow in gpm and is the diameter of thestack in inches. Since this equation is based, inpart, on the approximate expression for givenbyeq (42) and since, for a given pipe, the error fromusing eq (42) increases with (o r r), it is intended primarily for use where is not greaterthan 1 /3. However, analysis of eq s (41), (42),and (43) indicates that, even where the pipe isflowing full, eq (53) should give flow-rate valuesonly about eight percent less than if eq (43) hadbeen used in its development. This conclusion isconsistent with the finding that flow-rate valuescomputed for cast-iron soil pipe from eq (53) forr s =1.0 actually are from five to ten percent lessthan the values computed for very rough pipesflowing full at terminal velocity as indicated byfigure 2, based on Hunter's experiments.

The principal cause of air movement in buildingdrainage systems is the friction developed betweenthe high-velocity layer of water flowing down thedrainage stack and the core of air which it encloses. The friction thus developed drags airalong with the water. Under ideal conditions ofannular flow of water at terminal velocity wherethere is a positive pneumatic-pressure gradient inthe direction of flow, it is obvious that,, the air velocity may approach,but cannot exceed, the velocity of the water.However, general considerations indicate that thevelocity gradient in the water section is muchsteeper than that in the air core. Thus, it isbelieved that the mean velocity of the air core canbe greater than the mean velocity of the waterstream. Since available data do not establishthis relationship with acceptable precision, it isassumed here that the mean velocity of the aircore cannot be greater than 1 .5 times the meanterminal velocity of the water stream.The analysis of air flow in a drainage stackbegins by expressing the relation between air flowand water flow as

(54)Since is identical to in eq (53), will bereplaced by Next, will be replaced by theright-hand member of eq (53). This yields theequation

&=27.8(r.)>'(l-rf ) (55)in which is in gpm and is in inches. If^1.5, the equation becomes


24/58

4 6 10 2 A 6 100 2 46IOOO 2 460000 2FIGURE 15.

I .6

1.2

0.60.90 0.95 1.00 1.05

0/0N1.15

FIGURE 1 6.

^041.7(7%) (1 . (56)Figure 15 has been prepared from eq s (53) and(56). This figure shows the air flow computed forvarious flows of water in drainage stacks from 2

to 15 in. in diam. Nominal diameters of stackhave been used in the computations. Figure 1 6may be used in conjunction with figure 1 5, or withany flow-capacity equation based on the Manningformula, to estimate the effect on capacity odiameters which differ to a slight extent from thenominal values. If represents rate of discharge associated with nominal diameterand if represents rate of discharge associated/ \/ 3with actual diameter and if ) > then

In most plumbing codes a loading table for ventsis provided. The purpose of such a table is togive the information necessary to design the ventstack for the delivery of the amount of air requiredfor the control of pneumatic pressures at criticalpoints in the drainage system within limits of +1to 1 in. of water column from atmospheric. Ifthis range of pressure can be maintained, theeffects of pneumatic-pressure fluctuations on thefixture-trap seals will be negligible. The dimensions of pipes required to deliver given quantitiesof air at a pressure drop of 1 in . of water columncan be computed from the Darcy-Weisbachformula combined with the conventional formulafor expressing losses other than those associatedwith flow in long, straight pipes. This can beexpressed as

-I r~ / 7- * \ 5 ^ n / n r i (58)1 2In eq (58), is in feet, Dc is in inches, andis in gallons per minute; and the term is thealgebraic sum of the loss coefficients associated

with the entrance and exit conditions and withchanges in section, direction, etc. A temperatureof 60 F has been assumed in developing eq (58)If only the pipe-friction loss is considered, whichmay usually be done without serious error in practical situations, the equation

is obtained./2200X (A) (59)

Tables 6 and 7 show the range of conditions investigated by us e of the simulated stack shown in

figure 4. The individual values in this series ofmeasurements are shown in tables A-l and A-2 ofthe appendix.The experimental data afforded the informationnecessary for computing the terms andappearing in eqs (19) and (20). Next, thevalues of were computed from eqs (22) and (23).


25/58

TA BLE 6. TA BLE 7.

1

Branchdiameter

1.611.611.611.611.611.611.611.611.611.611.611.611.6 11.611.611.611.611.611.611.611.611.6 11.612.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.073.073.073.073.073.073.073.07

2

Thickness ofwaterlayer on wallof stack

0.16.16.16.16.16.16.16.16.35.35.35.35.35.35.35.35.58.58.58.58.58.58.58.16.16.16.16.16.16.16.16.35.35.35.35,35.35.35.35.58.58.58.58.58.58.58.16.16.35.35.35.58.58

3

Number ofbranchesflowing

11112222111122221111222111122221111222211112221111111

4

RateofflOW ainstack

4054809840548098569813718 0569813718 09815820 325 29815820 34054809840548098569813 718 0569813718 09815 820 32529820 325 280989813 718 0

20 325 2

5Ratio ofstackvelocity toco mputedterminal velocity

1.041.4 12 .0 82.551.041.412.082.550.47.821.141.500.47.821.141.500.40.64.821.020.40.64.821.041.412.082.551.041.412.082.550.47.821.141.500.47.821.141.500.40.64.821.020.40.821.022.082.550.821.141.500-821.02

6

Range of flowrates In eachbranch

15. 0 to 33. 213. 1 to 40 . 39. 4 to 26 . 58. 8 to 17. 417. 4 to 27 . 27. 9 to 29. 28. 2 to 20 . 68. 2 to 13. 817.9 to 32.28. 1 to 23. 48.4 to 14.48. 1 to 9. 7

8. 6 to 24. 28. 8 to 24. 89. 4 to 13. 79.28. 1 to 29 . 811.6 to 20.28. 1 to 13. 18.1 to 14.5

10 . 6 to 16. 811.0 to 16.611.130,020 . 8 to 34 . 123. 0 to 33 . 619. 0 to 24 . 2

15. 1 to 38 . 530.415. 8 to 30 . 819. 2 to 28 . 916.3 to 40.820 . 4 to 59 . 621. 2 to 32. 320 . 1 to 21 . 4

17.819. 0 to 28 . 217. 4 to 24 . 520.523 . 8 to 49 . 124 . 3 to 42 . 315. 4 to 25 . 716. 3 to 17. 8

16.918.617.452 . 1 to 128. 066 . 0 to 110 . 380 . 0 to 81. 042 . 0 to 149 . 049. 0 to 71. 153. 1 to 92 . 747. 8 to 53. 4

7

Range of headlo sses in branch(h 2 -hO

2. 57 to 3. 942. 94 to 4. 694. 44 to 6. 0 05. 44 to 6. 382. 59 to 3. 251. 94 to 4. 563. 25 to 5. 544. 44 to 6. 192. 94 to 3. 692. 88 to 5. 944. 44 to 5. 965.961.62 to 3.293. 0 6 to 5. 944. 44 to 5. 945.942. 0 7 to 4. 444. 0 6 to 6. 124. 44 to 5. 945. 94 to 7. 942. 69 to 3. 444.44 to 5.945.94

3.123. 37 to 4. 625. 56 to 7. 126.31 to 7.491.99 to 3.063.934.34 to 6.626. 37 to 8. 312.06 to 3.254.06 to 7.066.06 to 7.817.81

2.193.81 to 4.815. 31 to 6. 697.993. 43 to 4. 435. 81 to 7. 695. 81 to 7. 817.81

2.316.317.814. 46 to 6. 466. 46 to 8. 46

4.464. 46 to 8. 466. 46 to 8. 466. 46 to 8. 467. 71 to 8. 46

Above the stack-branch junction.

The values of thus obtained were plottedlogarithmically against This gave a seriesof straight lines with a slope of approximately 3/8.Next, from this plot, values of were pottedlogarithmically against for severalselected values of Again, this procedure

1

Branchdiameter

1.611.6 11.611.611.611.6 11.611.611.611.611.611.611.611.611.611.611.611.611.611.611.611.611.611.612.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.072.073.073.073.073.073.073.073.073.073.073.07

2

Thicknes s ofwaterlayer on w allof stack

0.16.16.16.16.16.16.16.16.35.35.35.35.35.35.35.35.58.58.58.58.58.58.58.58.16.16.16.16.16.16.35.35.35.35.35.35.58.58.58.58.58.58.58.16.16.16.35.35.35.35.58.58.58

3

Number ofbranchesflo wing

111122221111222o1111222211122211122211112221121112111

4

Rate offlow instack

4054809840548098569813718 0569813718 09815820 325 29815 820 325 25480985480989813 718 09813718 09815 820 323 2

15820 323 28098989813718 0

18015 820 322 7

5Ratio ofstackvelocity to co mputedtermina l velocity

1.0 41.412.082. 551.0 41.412.082.550.47.821.141.500.47.821.14l.EO0.40.64.821.020.40.64.821.021.4 12.082. 551.412.082.550.821.141.500.821.141.500.40.64.82.94.64.82.94

2.082.552.550.821.141.501.500.64.82.92

6

Ranpe of flowrates in eachbranch

26 . 8 to 34 . 419. 2 to 50.010. 7 to 50.010 . 9 to33. 425 . 0 to 38. 921 . 8 to 53 . G11. 4 to42.08. 0 to 32. 126. 6 to 26 . 712. 9 to 44 . 110. 4 to 26. 78. 8 to 16. 425 . 4 to 38 . 313. 8 to 44 . 913. 2 to 29 . 710 . 2 to 16. 715.0 to 53. 511. 7 to 29 . 311. 9 to 19.210 . 3 to 11.714. 5 to 41 . 311. 8 to 29.07. 9 to 18. 411.739. 7 to 64. 421 . 1 to 73 . 121. 0 to 50. 946. 4 to 6 G . 620 . 6 to 75. 622 . 8 to 53. 025. 1 to 72 . 721. 8 to 62 . 522. 4 to 39 . 631. 8 to48 . 816. 8 to59. 814. 8 to 35 . 748 . 3 to 10 9. 221.0 to 72.718. 9 to 49.519. 3 to 39 . 9

27.217. 3 to 26 . 619. 8 to 35.796 . 4 to 199. 577. 5 to 165. 1

83.5115. 1 to117. 271. 1 to 161. 050 . 3 to 111. 9

54.894.0 to 142. 264. 6 to125. 854 . 5 to107. 9

7

Range of headlosses in branch(Aa-Ai)

3. 643. 64 to 8. 143. 04 to 9. 144. 64 to 9. 223. 64 to 5.223. 64 to 8.143. 64 to 8.223. 64 to 8. 22

3.393. 39 to 7 . 724. 64 to 7.725. 59 to 7. 723. 39 to 5. 0 93. 39 to 7.644. 64 to 7. 645. 64 to 7. 643. 39 to 7.644. 64 to 7.645. 96 to 7.727. 59 to 7.643. 39 to 6.144. 64 to 7.644. 64 to 7 . 727. 59 to 7. 644. 29 to 5.924. 29 to 9. 365. 54 to 9.364. 29 to 5. 544. 29 to 9. 125. 54 to 9.184. 29 to 7.625. 54 to 9. 497. 49 to 9.244. 29 to 5.544. 29 to 9.185. 54 to 9.124. 29 to 9.044. 29 to 9.245. 54 to 9. 246. 29 to 9.24

4.294. 29 to 5.806. 29 to 9.125.61 to 10.115. 61 to 10 . 68

5.615.6161 to 10. 815. 61 to 10. 815.61

5. 61 to10. 365. 61 to 10. 365. 61 to 10. 36

Above the stack-branch junction.

gave a series of straight lines having a slope ofapproximately 3/8. Therefore, a value of a=3/8was adopted. Values of were then computedfrom eq (2 1) inserting the value of 3/8 for a.Evidently, the same value of applies approxi-21


26/58

mately whether the flow is from one or from bothbranches. Average values of (7 "=17.1 for thesanitary-tee fittings and =9.65 for the long-turn T-Y fittings were obtained. Hence, theequations representing flow conditions at the junction of the horizontal and vertical streams of thesimulated stack become, for sanitary-tee fittings,

F=8.33-2.48 * 7 F0.3 (62

6 / 8 3 / sF-8.33-17.1 g F3 / 8 J * T 5 / 8 - (60)andfor long-turn T-Y fittings( T\ 5 / 8 / 7 7 \ 3 / 85") ( 6 1 )

Equations (60) and (61) can be solved for corresponding values of and for any selectedvalue of [(T/Z)2 ) 5 / 8 ./A/A)3 / 8l This was done foral l the conditions investigated in order to determine the agreement between the experimentaldata and eq s (60) and (6 1). Representative curvescomputed in thisway have been drawn in figures17 and 18 . The agreement obtained indicatesthat the simplifying assumptions made in theanalysis are justifiedby the results obtained.

Thereare certain restrictions on the applicationof eqs (60) and (6 1). First, the head-loss termappearing in the denominator of and (see eq s(14) and (18)) represents the head loss betweenthe stack and a point in the horizontal branch 4pipe diameters from the stack. Second, the equations are based on a value of/=0-03 for the horizontal branches, and on a value of f=0.04 7 for thebend in a long-turn T-Y drainage fitting. Third,the data on which eq s (6 0) and (6 1) are based wereobtained for a 3-in. stack only. Although theanalysis indicates the equations should apply tostack diameters other than 3 in . this has not yetbeen demonstrated experimentally.

The data on single-branch flow in a prototypalstack are given in table 8 . These data were obtained by use ot the test systemshown in figures5 and 6.The experimental data shown in table 8 provided the information necessary forcomputing theterms and Fin eq (30). Next, values of < 5 werecomputed fromeq (32). The values of 5 thus obtained were plotted logarithmically againstThe data from both diameters of branch plottedfairly well along a si ngle line with a slope of approximately 0 . 7 , hence this value was adopted forthe exponent Then, thevalues of were co mputed from eq (31), inserting the value 0 .7 for a. An average value of (7=2.48 was obtained in thisway. Hence, the equation representing flowconditions at the junction of the horizontal andvertical streams of the prototypal stack usingsanitary-tee stack fittings becomes

Equation (62) can be solved for values ofcorresponding to selected values of This hasbeen done and the resulting curve is shown infigure 19 . The experimental data shown in table8 have been plotted in this figure. The agreemenbetween the curve and the plotted points issatisfactory.There are certain restrictions on the applicationof eq (62). First, the equation applies only whensanitary-tee stack fittings are used and when thehead-loss term appearing in the denominator oand (se e eqs (14) and (18)) represents thehead loss between the stack and a point in thehorizontal branch 4 pipe diameters from the stackSecond, the equation is