UNITED STATES DEPARTMENT OF COMMERCE Jesse H. Jones, Secretary

NATIONAL BUREAU OF STANDARDS Lyman J. Briggs, Director

BUILDING MATERIALS

and STRUCTURESREPORT BMS65

lVlethods of Estimating Loads in Plumbing Systems

by

ROY B. HUNTER

I S SUE D D E C E M B E R 1 6, 19 4 0

The National Bureau of Standards is a fact-finding organization;

it does not "approve" any particular material or method of con·

struction. The technical findings in this series of reports are to

be construed accordingly.

UNITED STATES GOVERNMENT PRINTING OFFICE WASHINGTON

FOR SALF. BY THE SUPERINTENDENT OF DOCUMENTS, WASHINGTON, D. C. PRICE 10 CENTS

Foreword

For many years, differences in the plumbing regulations or plulllbin~-('ode n'q uire­monts in different localities have been a source of annoyance to building owneIS, builders,and pu blic health officials and have hindered the standardization, on a geneI'Dl or nationalscale, of plumbing materials and equipment, and of construction desi~n. The principalreasons for the existence of plumbing codes are the protection of the health of theoccupants of a building and of the inhabitants of the community or cit~Tin which thebuilding is located and the protection of property from dam:tge by water or Sl'wage.

The results sought by plumbing requirements and health regulations applying toplumbing are or should be the same for low-cost or expensive and for small or large

buildings, and may be covered by general regulations applying to all plumbing. Anysimplified or standardized form of plumbing construction which gives thesl' results­

that is, which complies with 11eCeSsarygeneral regulations pertaining to snnit.ation orhealth-cannot be prohibitp<l without violating the inherent rights of building ownersand of the public.

Because of the above, this n'port has been writt.en, as succl'l'din~ reports on plumb­ing in the Building Materials and Structures series will be written, first. from the stand­point of essential general requirements and second from that of permissibh' constructionwithin the general requirements-both general requirements and pl'rmissibll' constructionbeing bnsed on sound physical principles.

This report deals with one of the factors which must be considered in tlH' selectionof adequate yet economical sizl's of pipes for plumbing systems-namely, tlH' load tobe expected from a given number and kind of plumbing fixture·s. It. is ('xpeet('d tlta t itwill be followed by other reports .in the Building Materials and Struetun's seri('s d('nlingwith other aspects of plumbing problems.

'Where the exercise of judgment is necessary in the choice of the nUlllt'ricnl value offactors used in developing and illustrating the applicat.ion of the method of estimatingloads as described in this report, it will be understood that such numerienl vnlues, whennot the actual results of Bureau tests or experiments, represent the author's judgment in

regnrd to the most suitabh' factor to use in tlH' application of the method, and thn t t.hesean' not t.o be regnnh'd as standard values, unless later approH'd as such by a repre­sentative and authoritatin body.

LYM ..l.N .J. BRIGGS, Director.

[II]

Nlethods of Estimating Loads in Plumbing Systems

by

ROY B. HUNTER

CONTENTSPage

Foreword n nnn __ h IIr. Introduction n 1

II. Purposc n 1III. Definitions and symbols_h n n 2IV, Basis of selection of pipe sizes n 3V. Character of fio\\' in plumbing systems_ 3

1. Cases of loading 5Statement of the problenL 5Development of the probability functiOlL 6Interpretation of the probability func-

tion ~ ..__IX Proposcd uses of the probability func­

tion for making load estimates __..__1. Pcrtinent information obtainable

from the probability functiolL_

VI.VII,

VIII,7

7

7

PageIX, Proposed uses of the probability func­

tion for making load estimates-Con.2. Valucs of t, T, 7", and q.. ~_h 83. Prescntation of probability data 114. Derivation of fixture weights (fix-

ture units) n __ n 135. Rclative load weights for diffcrent

conditions of scrvice hh 166. Application of load chart and

wcight tablcn n 18X. Adcquacy of the mcthod 19

XI. Estimates of sewagc loads_ n __ n _ n __ 20XII. Discussion n 21

XIII. Referencesn nn __ nn 21

ABSTRACT

This report describes a method of estimating thedemand and se\\'age loads for which provision should bemade in designing plumbing systems in order that theservice may be satisfacton'. The characteristics offlow through a plumbing system and of the operation ofsupply valves and plumbing fixtures are described, andtheir influence on the method of estimating the load tobe expected is discussed. The relath'e load-producingvalues of different kinds of commonly used plumbingfixtures are analy·zcd, and a tablc is de\'elopcd givingrelati\'e load \veights in terms of a load factor called the"fixture unit." An estimate curve developed bymeans of the probability function is given, and its usein conjunction with the table of fixture units is illus­trated.

1. INTRODUCTION

Aside from the choicp of materials and equip­ment as regards quality, style, and quantity offixtures, simplification and standardization ofpiping layouts offer the principal opportunitiesfor reduct.ion in cost of plumbing systems.Simplification and standardization in thisrespect must comply with accepted healthregulations and minimum rc'quirem~nts forplum bing, which in tUl'Jl should be based onscientific principles. This is true whpther thebuilding is small and "low-cost" or large and

[ 1 ]

expensive. Few, if any, existing plumbingcodes are based entirely on scientific principles;and they do not in general permit the simplifi­cation of plumbing piping layouts that couldbe accomplished well within the requirementsfor sanitation, and in many cases with moresatisfactory operation than can be obtained byinstallations under existing codes.

This report deals with one of the factors onwhich minimum requirements should be based,the maximum load to be provided for in plumb­ing systems. Other reports are planned, deal­ing with the principles of water supply andwater distribution in buildings, the principlesof building drainage, and the principles ofventing.

II. PURPOSE

The purpose of this series of papers is tocollcct in an organized form the mass of infor­mation obtained by the author over a numberof years, beginning with the investigation in1921 of plumbing of small dwellings, and includ­ing the current research (1937-40) on plumbingfor low-cost housing, together with the resultsof intervening experiments related to plumbingrt'quirements, and to interprpf, the results ofthese investigations in a form suitable for direct

and practical application. It is hoped thatthis series of papers will supply the logicalanswer to many of the controversial questionspertaining to pipe sizes and design of plumbingconstruction.

III. DEFINITIONS AND SYMBOLS

A number of terms employed in the plumbingindustry, and a few that are now introducedfor the first time, are here defined in the sensein which they arc to be used in this and laterpapers of this series. Insofar as they areadequate and applicable, the definitions fromRecommended .Minimum Requirements forPlumbing [1]1 will be utilized.

The plumbing system of a building includesthe water-supply distributing pipes, the fixturesand fixture traps, the soil, waste, and ventpipes, the building drain and building sewer,and the storm-water drainage pipes, togetherwith their devices, appurtenances, and con­nections within or adjacent to the building.

The building main is the pipe from the streetwater main or other source of supply to thebuilding served.

The water-distributing system is the pIpmgby which the water is conducted from thebuilding main to its various places of use withinand adjacent to the building, consisting oflaterals, risers, and branches.

The sanitary system of a building is the pipingsystem, including soil, waste, and vent pipes,that conducts waste water and water-bornewastes from the plumbing fixtures to the streetsewer or other place of sewage disposal.

A plumbing fixture is any receptacle throughwhich waste water or water-borne waste is dis­charged into the sanitary system.

A drain is any pipe that carries waste wateror water-borne waste in either the storm orsanitary system.

A fixture drain is the single drain from afixture-trap outlet to its junction with anotherdrain.

A waste pipe is any drain that carries thewaste water from one or more fixtures otherthan water closets.

1 Figures in brackets indicate the literature references at the end ofthis report.

[ 2 ]

A soil pipe is any drain that carries the wastewater from one or more water closets with orwithout the waste water from other fixtures.

A stack is a vertical section of pipe extendingtwo or more stories with branches therefrom toserve the stories through which it passes, aswaste stack, soil stack, or vent stacIe

A horizontal branch is a drain, extendinglaterally from a sailor waste stack, with orwithout vertical sections, to which the fixturedrains in the same story are connected.

A building drain is that part of a sanitarysystem which receives the sewage from soiland waste stacks and conducts it to the buildingsewer.

A building sewer is the main trunk of a sani­tary, storm, or combined system from a point 5feet outside the inner face of the building wallto the street sewer.

A primary branch is the single drain line lead­ing from the base of sailor waste stack to itspoint of connection with the building drain oranother branch thereof.

A secondary branch is any branch of the build­ing drain other than a primary branch.

A vent or vent pipe is any pipe installed in asanitary system for the purpose of permitting acirculation of air through or within the system.

A vent stack or main vent is a vent pipe parallel­ing a sailor waste stack to which the branchvents from the stack and its horizontal branchesare connected.

Demand load is the volume rate of demand

for water imposed on the water-distributingsystem or any branch thereof by the use ofwater, as by plumbing fixtures.

Sewage load is the load in volume or volumerate imposed on the sanitary system or anybranch thereof by the use of plumbing fixtures.

Charging load is the volume inflow into thesanitary system from one or more plumbingfixtures, applicable when the capacity of thepipe receiving the load depends mainly on theinternal volume (cross section times length) ofthe pipe.

Receiving capacity is the internal volume ofa section (drain) of the sanitary system, appli­cable when the total sewage load is an additivefunction of the separate volumes but not of theseparate rates.

Terminal velocity is the constant velocity of

flow that exists under fixed conditions whenequilibrium between the forces (gravitational)producing and the forces (frictional) opposingflow has been established.

Design factor m is the particular value of rout of n fixtures that will be found in operationa selected fraction of the time under the as­sumed conditions of use.

Fixture unit, or load factor, is a numericalfactor which measures on some arbitrary scalethe load-producing effect of a single plumbingfixture of a given kind. The use of the fixtureunit makes it possible to reduce the load­producing characteristics to a common basis.

n= the total number of fixtures or supplyopenings of a given kind in the system.

r= the number of fixtures out of a total ofn which at any given instant of ob­servation are found operating to im­pose a demand load on the supplysystem, or a sewage load on the drain­age system.

m=the design factor (see definition above).q=the average volume rate of flow, in

gallons per minute, to or from aplumbing fixture during actual opera­tion.

Q=the total volume of water in gallons thatflows to or is discharged by a fixtureat each use.

t=the average duration of flow in secondsfor a given kind of fixture for one use.

T=the average time in seconds betweensuccessive operations of any givenfixture of a particular kind.

r=a time interval in seconds such that theevent in question (for example, ex­actly r fixtures will be found operat­ing) will occur for an aggregate of 1second.

C~=the number of combinations of n thingstaken r at a time.

p;'=the probability of exactly r fixtures outof a total of n fixtures being found inoperation at an arbitrary instant ofobservation.

r=n

:8p;=the probability that some number ofr=m fixtures between r=m and r·=n, in-

clusive, will Iw found operating at anarbitrary instant of observation.

IV. BASIS OF SELECTION OF PIPE SIZES

In order to select from a series of available orstock sizes the minimum size of pipe adequateto convey a given flow of a liquid, it is necessaryto know accurately the capacities of the differ­ent sizes of pipe for the conditions under whichthe pipe is to be used and to know accuratelythe load the pipe will be called upon to carry,capacity and load being expressed in the sameunits or in convertible units.

Pipe formulas, given in terms of the velocityof flow and factors on which the velocity de­pends, are the usual means employed in esti­mating capacities of pipes. It is an easy stepto convert a formula giving the velocity of flowin a pipe into one giving the volume rate of flow.As previously indicated, the applicability ofpipe formulas to the conditions existing inplumbing systems in service will be treated inother papers. However, since it is necessary toexpress capacity and load in the same units inselecting the proper size of pipe for a given use,it becomes necessary to consider the units inwhich both the expected load and the capacityof the pipes involved can be expressed beforechoosing the units in which load estimatesshould be expressed. A pipe-flow formula ismerely an equation of dynamic equilibrium and,as such, applies only to the particular condi­tions-namely, the condition of uniform con­tinuous flow in the pipe. Hence the conven­tional pipe formulas apply to the irregular andintermittent flows that occur in plumbing sys­tems only during that time (usually very short)and in that section of pipe in which the variablefactors involved (velocity or volume rate offlow, pressure or hydraulic gradient, and hy­draulic radius) arc constant.

V. CHARACTEI~ OF FLOW INPLUMBING SYSTEMS

Probably the best conception of the limita­tions of the applicability of a pipe formula foradequately estimating the capacities of plumb­ing pipes, and the best indication of the logicalunits to apply in expressing capacity and loadin selecting the proper size of pipe to use inparticular parts of the system, will be givenby considering the changing character of flow

[ 3 ]

as the water passes through the system fromthe source of supply to the building sewer.Water closets with flush-valve control of thesupply are chosen for this illustration of vary­ing flow characteristics, partly because criticalloading conditions in the various parts of theplumbing system are usually produced bythese fixtures and partly because the factorsthat determine the load for an individualfixture are more nearly constant and can bemore readily evaluated than for other fixtures.

Assume that each water closet demands for

each flush Q=4 gallons in the average timet=9 seconds, giving an average rate of supplyto the closet of approximatcly q=27 gpm, andthat each of these demands will occur on theaverage of once in T=5 minutes=300 seconds.Since the water-supply pipes are flowing fullunder pressure, there will be continuity of flowin both time and quantity in any part of thesupply system through which water flows tothe water closet or to a group of closets; andtherefore the peak load at any time will be thesum of the separate demands, r, occurring atthat time, or rq gpm, where q is the averageflow per fixture. In this case a pipe formulais applicable for computing limiting capacitieson the basis of diameters, lengths, roughness,and available head. The difficult part of theproblem is to estimate the maximum r out of atotal of n water closets supplied by the systemfor which provision should be made.

In the drainage system the flow from thewater closet through the fixture drain (closetbend) into a horizontal branch will be at aboutthe average rate of 27 gpm for a period of from8 to 10 seconds. However, the flow will beless nearly uniform than the flow into the bowl,rising to a rate offrom 30 to 40 gpm temporarily,as siphon or jet action comes into play in thebowl, and dropping immediately following thepeak siphon action to a rate approximatelyequal to that of the supply at the time.

In the horizontal branch the temporary peakflow will flatten out, and the stream will tendto assume the terminal velocity (see definition)for the diameter and slope of the drain and theexisting volume rate of flow. As the streamenters the vertical stack, the sudden change indirection pilcs the water up, tending to form aslug of water at the entrance. In fact, the

surge from a single water closet entering a 3­or 4-inch vertical stack through a short llOl'i­zontal branch of the same diameter will almostfill the stack at this point. Immediate accelera­tion tnJ~es place in the vertical stack and con­tinues throughout the length of the stack oruntil the terminal velocity of the stack isreached for the volume rate of flow.

From the stack the stream will enter thesloping primary branch at a velocity muchgreater than the terminal velocity for thatbranch (shooting flow) 2 and will be deceler­ated until the terminal velocity for that volumerate of flow is approximately reached. If thevolume rate of flow, the entrance velocity, andthe slope, diameter, and roughness of the pri­mary branch are in the proper relation, thl'transition from shooting flow at the entrancpto gravity flow will occur in a hydraulic jump,a phenomenon which frequently occurs in theprimary branches of building drains.

Throughout the drainage system in the direc­tion of flow there is a continuous decl"pasein the volume rate of flow from a single closetor a group of closets flushing at the same time,a corresponding increase in the time of flow,and a continuous decrease in the peaks pro­duced by siphon or jet action in the bowl orby the overlapping of the flows from differentclosets.

The conception of flow in plumbing drains asdescribed has been verified by numerous ex­periments in the laboratory and by numerousobservations and depth measurements made onthe building drains of large systems in actualservice. In plumbing drains the peak rate offlow from a water closet at the point of entranceto the horizontal branch is usually reduced toabout one-half that rate at the point of transi­tion from shooting flow to gravity flow in aprimary branch, and the volume rate of flowcontinues to decrease from that point on.

If a single water closet flushes on the averageat intervals of 5 minutes and delivers 4 gallonsfor each flush, the stream in the drain tendstoward a continuous flow of Q/T=4/5=0.8 gpmas the distance from the water closet increases.Likewise for a· very large number of water

2 Shooting flow h('r(~rdt'rs to the high-yelocity constantly ctect'leratingflow that occurs in the primary branch of a building drain betwecn itsjuncture with the soil stack and the hydrauIic jump in the primary

branch.

[4 ]

closets, n, eneh used nt thnt average rate, 1Ulda very long building drain, the flow tendstoward a uniform value 0.8 n gpm. If n isgreater than Tit, it is obvious that flow in themain building drain will be continuous for thegreater part of the time that each of the closetsis being used at the average rate of once in Tseconds, and that the tendency towat'd uniformcontinuous flow will increase as the size of theplumbing system increases-that is, as n 1tndthe length of the drains increase. Hence in avery large system in which all n closets are incontinuous use, the outflow will be continuousand the rate will fluetuate around 0.8 n gpm.

Since the stream in the main building drainand building sewer of large buildings approachesunifot'm flow, and since these drains are ordi­narily laid with a uniform slope, load andcapacity can be most adequately expressed involume rate of flow and a pipe formula foruniform flow may be used in estimating limitsof capacity for building drains. However, itshould be kept in mind that the volume rate offlow in any particular section of the drainagesystem is not an additive function of the separ­ate volume t'ates of flow into the system.

There is another consideration that has adirect beat'ing on the method chosen for esti­mating the loads to be provided for in a build­ing drn.inage system and on the choice of unitssuch that load and capacity may be expressed inthe same units. The horizontal branches, theentrance fittings to soil and waste stacks, andprimary branches arc paTts of the drainagesystem in which critical loading or possible over­loading arc most likely to occur. In horizontalbranches which lie near the fixtures, and to alesser degree in the stack fittings and hori­zonta.l branches, the determinate factors in theselection of adequate drain pipes are the charg­ing load and the receiving capacity. In esti­mating the charging load to he provided for,the problem is again to obtain a reasonableest,imate of the value of r to be provided for inrelation to n. The details of the method re­lating to l"Cceiving capacity will be comph,tedin the papers dealing with capacities.

1. CASES OJ<' LOADING

From the charaetet' of the flow in the water­supply system and in the drainage system as

[5 ]

described, it is obvious that there are threedistinct cases of loading to be considered:(1) One applying to the supply (the demandload) which will be measured by '1:,rq, where r

is the number of separate like demands at thetime for one kind of fixture and q the averagevolume rate of flow per demand, rq the totalflow for r fixtures of one kind, and '1:,rq the sumor total flow for fixtures of different kinds;(2) one applying to horizontal branches (thecharging load) which will be measured by '1:,rQ,

in which r is the number of like fixtures flowingat the time and Q is the average volume ofsewage pel' fixture introduced into the drainwithin the time necessary to consider in relationto a particular location of the drain; and (3) athird case applying as a limit for building drainsand building sewers which will be measured interms of J:.nQjT, in which n is the total numberof anyone kind of fixture installed in the system,Q the average volume used per fixture, and Tthe average time between uses. In cases 1 and2, the load rq to be expected and provided for inthe design of the plumbing system depends,with a few exceptions, on approximately thesame time factors, t and T, and hence bearsapproximately the same relation to the numberof fixtures, n. (See page 20.)

VI. STATEMENT OF THE PROBLKM

As stated in describing the conditions offlow in plumbing systems for which load esti­mates arc sought, the most difficult part of theproblem is to estimate the proportional part ofthe total possible load that should be providedfor-that is, the number of demand or charginglonds, r, bearing some logical relation to thetotal number of fixtures, n. In an earlierpaper [1] the authors suggested the relationbetween rand n expressed by a probabilityfunction as a possible means of estimatingloads to be expected in plumbing systems anddl'ydoppd the function applying to certainsJweia1 cases.

It now appears that the possibilities of theprnetical application of this function and thelimitations of the method under certain condi­tions have not been understood clearly. ItSl'l'ms ndvisnble, therefore, to consider theprobability function in its more general form,

to explain more fully the suggested method ofemploying the function in making the estimatesin question, and to point out certain limitationsin its practical application. The problemmay be stated as follows: Assuming that thereare n fixtures in a system, each operated oncein T seconds on the average, and that eachoperation is of t seconds' average duration,what is the probability that r fixtures will befound operating simultaneously at any arbi­trarily chosen instant of observation?

It is necessary to define the expression,"operating simultaneously," in order to com­pletely define a particular even t of "r fixturE'Soperating simultaneously." In the followingdevelopment of the theory, this event will beconsidered as occurring when r, and only r.fixtures are found flowing at tIlE' instant ofobservation; and hence the r fixtures foundflowing will include all those, and only those,which began their opera.tion during the t-secondinterval immediately preceding the instant ofobservation.

VII. DEVELOYtdENT OF THE PROB­ABILITY FUNCTION

By the gem'rally accepted concept of prob­ability, the probability that a particular fix­ture out of a number, n, will be found operatingat any arbitrarily chosen instant of observationis tiT, where t has been defined as the durationof each operation and T as the time betweenoperations of each fixture. In the same man­ner, the probability that the particular fixturewill not be in operation at the instant of obsE'r­vation is I-tiT 01' (T-t)IT.

A law of combinations that applies to tllE'composite event of which the probability issought in this problem may be stated as fol­lows: The number of ways in which two ormore independent events can occur together isthe product of the ways each can occur sepa­rately. A similar law of probability may bestated as follows: The probability of two ormore independent events occurring together, inthis case at the same instant, is the product of[,he probabilities of their separate occurrence.

By the law of combinations, the probabilitythat none of the remaining n-l fixtures willbe operating at the instant of observation is

(T-t)"-l. The probability that a particularT

one of nfi~tures and none of the remaining n-lfixturcs will be operating at the' instant is

But since there an' 11 fixtun's, there an' n waysin which the event, am' and only one fixturefound flowing, can occur, and hence the proba­bility of this event l)('ing found occurring atthe instant of observation is

(1)

This can be generalized for any group of r,and I' only, constituting the particular eventfor which the probnhlity is sought, by againapplying the definitions and laws.

In any group of I' fixtures the probabilitythat any particular OIll' will begin to operatewithin a given t seconds is tiT, and sinee thereare r cases in thp group, the probability, thatall r fixtures will begin opC'ration within thesame t seconds is (tiT)'. Also, of the n fix­tures, anyone of

n(n-l) (n-2) to r factors_ lIJ __ a:-r!------····. -"!(n-r)!- ·r

different groups is equally likely, and hence bythe law of combinations the probability thatsome one of the equally likely groups of r fixtureswill be in operation at tIll' instant of observa-

tion will be O{~)r. As l)('fore. the proba­

bility thnt tlH' remaining 11 -I' flmn will fall

(T-t)"-routside' this particular t s('conds is -r .Therefore, the probability that exactly r fix­tures will be operating at n particlllnr instantof observation in the mann('r defined is

This is the general term of a series, expressingthe probability that r out of n fixtures will 1)('found flowing at any a.rbitl'Hr'y instant ofobservation. This spries may 1)(' representedas

[6 ]

or

Kquation :3 is ('quivalent to the binomial

(6)n-I' t

=r+1XT-(

IX. PROPOSED USES OF THE Pl{OB­ABILITY FUNCTION FOR MAKINGLOAD ESTIMATES

There are certain characteristics of tll<'prohabilit.y series given in eq 2 which will be ofmaterial aid in determining the conclitions onwhich the design factor m is based. In theseries, eq 2, the vnhw of successive terms in­creases as I' increases up to the most probable

It ma.y be helpful in j lIdging the reasonable­ness of the proposed application of the proba­bility function to summarize the informationobtainable from the probability function beforeproceeding with the select.ion of the tilll<'factors applicable to practical cases.

The following per·t.inent information can 1)('

obtained from the equations developed:(a) The probability that a given number of

fixtures, m, out of a total of 11, will be operatingat an a.rbitrary instant of observation, deter­mined by eq 2.

(b) The fraction of the time that m and onlym fixtures will be operating at the same instant,a second interpretation of eq 2.

(c) The fraction of the total time that anynumber of fixtures greater than the designnumber, m, will be operating at the sameinstant, determined by eq 5.

(d) The ratio of any two successive terms intIll' series of eq 2; for example, the ratio of therrHct.ion of the time 7'+] 1.0 tIll' frHctioll of t.hetill}(~ I' fixtures will )(' operating' at. t.he SlUIl!'

instant, determined hy

1. PERTINENT INFOIU1ATION OBTAIN ABLE FROM

THE PROBABILITY FUNCTION

Equations 4 and 5 are based on the assump­tion that all n fixtures will be in con tin \lO\lSuse over the entire time 7 at the average rateof once in T seconds. The time 7 may bereduced to days or years on the basis of thedaily period of peak use, by assuming or de­termining this period and comp\lting on thebasis of a day of that length.

(5)

(3)

IS theand

r=n

'L,]>';= 1/7 ...r=m+l

'fj]J"= ,£O"tr (T- t) n--r = 1r=O r r=O r 1'n

time of r fixtures operatingtotal time

3 Equations 2 and 3 may he developed by other rncthods and in dif·ferent. forms. However, the results will be identical or may be reducedto an identity, as cyidenced by the function denlopcd by Fry [2] for theanalogolls prohlcm or husy lines in telephone fiysteIl1s (see eq 156, p. :~:{;\of thc referencc), which differs from eq 2 only in the form to which it isreduced and in the notation employed,

is also the probability of the occurrence.Hence for any given values of nand r, if theprob!lbility function as developed is multipliedby a time 7 in seconds such that

(t)'(T-t)7p';=70'; l' -----rr' ,,-r=l second, ... (4)

VIII. INTERPRETATION OF THE PROB­ABILITY FUNCTION

. f (t + 1'-t)" 1(\XpanSlOn 0 l' -1' - =.

As previously stated, the probability funetion,p;', eq 2, gives the probability that exactly rfixtures out of a total of n will be found operat­ing at an arbitrary instant of observation, pro­videel that all n fixtures are in continuous usea t the assumed rate. The pro babili ty, p~,may also be interpreted as the percentage orfrnction of the time in the long l'tm that r flowswill occur in the manner defined, since thefraction

('onven tional ('xpn'ssion for certainty, smceeither no fixtun~s, 1'=0, or some number offixtures from 1'= 1 to r=/1 must be operatingat, the instant of observation.3

(·lIe N]uat,ion sign i(ie" Lilat /' fixtures will he in"illlult.alwo\ls operation for an aggregate or Isecond out qf every 7 seconds that all n fixturesare in usc at the assumed rate. Likewise, thecondition that a chosen design factor, r=m,will not be exceeded more than a given fractionof the time 1/7, is expressed by

[7 ]

value or values and then decreases. Also, theratio of any term to the preceding term of tht>series decreases continuously as I' increasesfrom 1'=1 to r=n. These characteristics makeit possible to determine the value m that willnot be exceeded more than the fraction l/T ofthe time without completing the entire summa­tion from the m+ 1 to nth term.

If p~,<l/T, and if the ratio (eq G)

n-m t 11'+1 XT-t<'j'

T=l1

the slUlllnation ~p'; must bc less thaI! 1fT,1=111+1

because the finite series

{~+~(2~O)+~(2~O)(2~O/)+ .. }

which results fmm the scries

P::'+l+P::'H+ ... +p~,

by multiplying and dividing this last series byp;;', substituting the value 1/2 for the ratioP::'+l/P::', and replacing the factor P;;' by l/T, isless, term by term, than the infinite series

which is known to approach the value l/T asthe number of terms increases without limit.When the series satisfies this condition, it canbe positively stated, without summation, thata design factor r=m will be exceeded less thanl/T of the time.

If n-r X t > /') ..1'+ 1 T-t 1 -, summatlOJI IS necessary

in order to determine a value of r=m that willnot be exceeded more than l/T of the time, butit is unnecessary to carry the summation beyoml

n-r t

the term when' 1'+1XT_t<1/2.

2. VALUES OF t, T, T •. \XD q

In applying the probability function forestimating the design load mq, it is necessary tosclect values of t, T, and T from which to com­pute the value of m and to soled a value of q.the factors excepting T pertaining to a particula rkind of fixture and service. The actual values

[8 ]

selected in any case are largely a matter ofengineering j uclgment. In this connection, itis to be understood that in the followingdevelopment and illustrative examples, thevalues selected represent the author's judgmentin regard to the appropriate values for pro­ducing satisfactory service and are based onthe author's interpretation of the informationavailable.

For the purposes of this discussion satisfactoryservice is defined in a relative sense as that inwhich interruption in service because of con­trollable factors, such as the sizes and arrange­ment of pipes, is infrequent and is of sufficientlyshort duration to cause no inconvenience inthe use of fixtures or any unsanitary conditionin the plumbing system. Attainment of satis­factory service will depend on the selection ofthe design factor m, or more specifically on thevalue of T from which the value of m is com­puted. The value of T selected for illustratingthe proposed application is 100 seconds, whichprovides for wholly satisfactory service 99 per­cent of the time and for reasonably satisfactoryservice all of the time if the design load mq gpmis not greatly exceeded. In this connection,it will be observed that if m is exceeded inactual service it is most likely to be exceededby one fixture only and is progressively lesslikely to be exceeded by two, by three, or more.

Obviously t and q bear a direct relation to Q

in respect to the values necessary to providesatisfactory service if m fixtures arc in opera­tion simultaneously. Since there is a consider­able range in the values of t and Q on whichthe value of q depends for any particular fixture,it will be very helpful to the engineer in deter­mining reasonable values to be used for aparticular kind of fixture to consider the charac­teristics of operation of that kind of fixture.

It is a characteristic of water closets thatthey will operate more or less effectively underany average rate of supply from about 15 gpmup to rates of about 30 gpm or more deliveredin any time ranging from about 6 seconds up.For each type and design of water-closet bowlthere is an intermediate smaller range of aver­age rate of supply within which there is nod()tectabl(~ d ifJ'erpJ]cP in the efl'ectiveness ofthe flush in emptying the bowl of it.s content.s.There is likewise a l'angl' in time of flow within

.~--~~.~..~.~."--:----------------

246TIME - SECONDS

FIGURE ] .-Typical lime-supply curves for lOater-closclbowls.

flush valve or flush tank: it rises to a maximum

at or near the beginning and gradually de­creases as the valve closes or as the tankempties. The characteristics of an effectiveflush may be summarized as follows:

A quick priming of the siphon; a continuoussiphon action for sufficient time to clear thebowl of its contents and carry them throughthe trapway; and a breaking of the siphonaction before the flow ceases, in order to refillthe trap. Whether by design or chance, thecharacteristic operation of conventional supplydevices (flush valves and flush tanks), if pro­perly adjusted for volume and time, controlsthe rate of supply in a manner that meetsthese flushing requirements admirably. Infigure 1, curve 1 is the reproduction fromCamp's report of a typical rate curve for itflush valve, representing it flow of 3.8 gallonsin 8.5 seconds giving it peak rate at the end oftwo seconds of approximately 37 gpm and anaverage rate of 26.9 gpm over the time thevalve is in operation. Curve 2 is the ratecurve for a flush tank plotted from Golden'sdata for a flow of 4 gallons in 9.1 seconds givingit peak rate about 31.5 gpm at the end of 1

second and an average rate of 26.4 gpm. Inboth cases the flow reaches a rate within thefirst second amply sufficient to prime the siphonof any standard type of closet bowl and there­after is maintained for 6 or 7 seconds withinthe range required to produce continuous

which thc bowl will be effectively emptied ofits contents with an average rate of supplyanywhere within the smaller range referred to.

From the evidence of experiments reportedin the earlier paper [1], the Subcommittee onPlumbing of the United States Department ofCommerce Building Code Committee agreedupon an average rate of 30 gpm for 10 secondsas a reasonable and safe basis for estimatingloads to be expected in building drainagesystems. The experiments referred to weredesigned to obtain the maximum loads percloset that could be delivered to the drainswithin the operational range of water closets,and no attempt was made to determine eitherthe most effective rate of supply for a particulartype of bowl or an average rate that would pro­duce a satisfactory flush in all types of closetbowls. It is to be expected, as has been thecase, that overestimates would result when themaximum values of all load fa(~tors involvedare employed in estimating. More recent ex­periments [3] by Camp give rates of supply forsafe and economical flushes ranging from 20gpm to 29 gpm for different types of closetbowls and times of flush ranging from 7.5 to 9seconds, omitting data for one bowl with spiralflow action. The averages for six differentbowls are 25.9 gpm and 8.2 seconds. Unpub­lished data from still more recent experiments[4] at the National Bureau of Standards indi­cate that the most effective removal of thecontents of a closet bowl occurs with rates ofsupply ranging from about 20 to 24 gpm andthat the complete contents of the bowl areremoved, if removable by any rate of flow, intimes ranging from about 6 to about 10 secondsfor different types of bowls.

In both cases the data were obtained withapproximately uniform flow through the bowl,and that average rate of flow which maintainedcontinuous siphon action through the bowl wastaken as the criterion for the most effectiveflushing rate. Considering the wide ranges inrate and time of supply within which a closetbowl will operate effectively, Camp's andGolden's data may be regarded as in sub­stantial agreement. However, none of theexperiments cited closely simulates serviceconditions, for the rate of supply in actualservice is not uniform, whether supplied by

r 9]

40

~& 30,wC)a:<t

i3 20(/)oLLo

~ 10<ta:

oo

I. FLUSH VALVE2. FLUSH TANK

8

\'-

- ._-- ---- ------------TARLE l.--Vallles of tiT, q. and Q

to the overflow. Ordinarily only a fraction,possible ~, to ~~,of these volumes will be used fora bath. The time, T, between uses will includethe time required to draw the water, the timetaken for bathing, the time required to emptythe tub, and any additional time consumed inthe complete bathing operation. This totnltime, T, between operations of the fixture seemslikely to range from about 15 minutes for hur­ried baths to about 30 minutes for ba,ths takenmore leisurely. It also seems likely thatsmaller quantities of water will be used underthe former than under the latter conditions.Now, assume that 8 gpm is an ample averagprate of supply for a bathtub. This is GO percenthigher than some [5] and 20 percent lower thanother [6] estimates of a satisfa,ctory average ratpof supply for a bathtub. An average rate ofsupply of 8 gpm, an assumed average time offilling of t=60 seconds, and an assumed timebetween operations of 1'=900 seconds (15minutes) would provide for an average of 8gallons per bath. For an average volume of 16gallons per bath and the same rate of supply. 8gpm, 120 seconds (t) will he required to drawthe bath. If the rate of operation (averagptime taken per bath) is once in ~o minutes, giv­ing 1'= 1,800 seconds, the ratio of tj T is thesame in both cases; 60j900= 120j1.800= 1j1.5.Since for a given value of n the value of p; forany value of r is determined by the ratio tjT, theprobability of a selected design load mq beingexceeded will be exactly the same for the twocases cited or for a,ny other case in which thetime l' is proportional to the volume used andthe same basic rate q is employed. From theseconsiderations, a design load mq, for bathtubsin congested scrvice, based on an average rateof supply of 8 gpm, and a ratio tjT= 1/15 ap­pears to insure fairly satisfactory service and isused as the basis of the development, and com­parisons following.

'1'11(' values selected for the three fixtlJreS dis­cussed are given in table 1.

siphon action. The total volume, the averagerate of flow, and the duration of the flush inthese examples are all greater than necessaryfor an effective flush in the types of bowls mostcommonly used-washdown, reverse trap, andsiphon-jet bowls. It is impractical to attemptto estimate individual variations either inrates of supply or in rates of use of fixtures.Hence, it is advisable to set the factors chosenfor making estimates high enough to allowsome leeway in adjusting down to the volumeand rate of supply needed for satisfactoryservice in particular cases, since it is impossibleto adjust for greater rates of supply than thesupply pipes are capable of delivering.

Considering the problem of estimates fromall angles, an average rate of supply of 27 gpmfor a duration of 9 seconds, giving a volume ofapproximately 4 gallons per flush, appears tomeet requirements generally as nearly as canbe done in l'Ound numbers and will be employedfor flush valves for water closets in evaluatingthe probability function in the further develop­ment of the methods of estimating demandloads.

The rate of supply to flush tanks for wa1<'rclosets is not related direetly to the rate ofsupply required by the closet for effectiveoperation. The only essential for operationof the fixture is that the tank refill in the interimbetween operations. A rate of 4 gpm isordinarily considered sufficient, which for avolume of 4 gallons gives a value of 60 secondsfor t. The value of T should obviously be thesame for flush-valve and flush-tank supply.

In case of faucet-controlled supply for fix­tures, it is not possible to basc the time factorsor the quantity of water used on the character­istic operation of the fixture as was done forflush valves for water closets, because themanner in which faucets are operated dependslargely on personal habits or preferences.For such fixtures an arbitrary selection ofvalues, made from a consideration of relativerates of supply and volumes used, appears tobe the only recourse.

An avernge rate of supply of 8 gpm would per­mit the drawing of 8 gallons in 1 minute (60seconds), 16 gallons in 2 minutes, 24 gallons in~ minutes. etc. Bathtubs. depending on sizeand style, hold from 25 to 40 gallons when filled

Kind of fixture

Flush valves for \'rater closf'ts . __"Flush tanks for wftt(~r closets _TIathtnhs a_u u __

r 10]

WI'

9/300

60/300 I1/10

q in gpm

2748

44

8to W

Equations 4 and 5 arc not satisfied in generalby integral values of rand n and a given valueof l/T; that is, for occurrence exactly a givenpercentage of the time. However, by trial anderrol', a value of t for each given value of ncan be computed for which the probability ofoccurrence is in excess of a given perccntage,for example 1 percent of the time, by the small­est possible fraction. From this probability,71;, a particular value m of r, for which theprobability of occurrence of any numberg'renter than m is not greater than a givenpef'(~(mtagc of the time, may be rcadily deter-

t=nmined by an approximation of 'i:.p~.

t=m+1If P,::+l < (1/2)p;:', no further computation IS

necessary as pointed out on page 8. If71;;,+1 > (1/2)p;':, it will ordinarily be necessaryto carry the summation to a few terms only todetermine the value of m required.

Probability data for flush valves for waterclosets, flush tanks for water closets, and bath

.,v. PHESEN'l'A'l'ION OF PROBABILITY DATA tubs are given in table 2, in which t is thegreatest number of fixtures out of n that willbe in operation 1 percent of the time and mis the number of fixtures that will not be ex­ceeded more than 1 percent of the time.

TARJ,E 2.-Probability data for three types of fixturcs

Flush valves, forFlush tanks, forBathtubs, for

I/T~9/3OO~3/100I/T~f.o/300~ 1/5I/T=I/15

n

r V;mnrV;mnrV;

III-- ----------------------6

20.0119 232O.0960 2320.0124 2

Hi3.0102353.0512 383.0118 :J

304.0101474.0281 4154.012ti 4

475.0104595.0165 5225.0107 5

666.01066157.0138 7316.0118 G

857.010072510.011810407.0112 7

1078.010584014.011514599.0104 9

1299.010496420.01122011314.010214

15110.01011013236.01043717219.010319

19912.010212240GO.0101G722123.010223

2991G.01011t130574.01027830!)30.010031

It will be observed that for flush valves tiTis small, and r=m for all values of n in therange given, and that the greater tiT the smal­ler is the range in which r=m.

Figure 2 shows the curves obtained by plot­ting m and n for the three fixtures, curve 1 being

100

I. FLUSHVALV ES, y/'f =o. 0 I

2. BATH TUBS, yI'f= 0.013. FLUSH TANKS, V1=0.01

./

4. FLUSH VALVES, y/1=0.000001

./.../

./

./

./,/

V .../..././..,/'.,.

/V/'

---- --,/'

..--

./4

,/'-- ~ I

l.?~ .----'

..--- - - -

75

E 50

25

oo 50 100 150

n200 250 300

FIGUUE 2.-Probable relation of In to n from table 1.

[ 11 ]

500

I. FLUSH VALVES /'2. BATH TUBS

/",V3. FLU S H TAN K S

/'"

./

4. AVERAGE FLOW, FLUSH VALVESVAND

FLUSH TANKS V5. AVERAGE FLOW

BATH TUBS.//'"

V/'"./

/'"

/' V/V ........-

V./ 1/ ........-

V

..//':::::::::

/'L.--:::

./

/"' V..---:;::::;:....--

V/'......--~.-/.-,/ /'"V---

..•......../ ..--

/ 3/' V..•.......

........-

V ---V/'"2'..•.......

......--.-V -f-"""./

---/'

......--4..•••...••.-/

----./

......--

//~.V•...••

........-

----I--

/~/' .-/5---V ---/ "..-.-/

........- -----:P

........-

vc.---

~?-:..--

400

300

200

100

oo 50 100 150

n200 250 300

FIGURE 3.-Probable flow in relation to n.

for flush valves, curve 2 for bathtubs, andcurve 3 for flush tanks. A value of m for flushvalves that would not be exceeded for morethan one-millionth of the time was also com­puted, and the results are represented in figure2 by curve 4.

Estimates of pipe capacities will ordinarilybe obtained in flow units, for example in gallonspel' minute. Hence it will be more convenientin practice to have the demand load expressedin the same units. This may be obtained byplotting mq and n as shown by figure 3, in whichq is taken as 27 gpm for flush valves, 4 gpm forfiush tanks, and 8 gpm for bathtubs. Theaverage flow, nQ/T, during the time the fixturesare being used at the rate of once in the time Tis also represented in figure 3, for flush valvesand flush tanks by line 4 and for bathtubs byline 5.

Curves 1. 2, and 3, in figure 3, show the I'ela-

tion of demand loads to number of fixtures.based on estimated time factors representingcongested conditions of service-that is, themaximum practical rate at which fixtures canbe used continuously in actual service. Assum­ing the COITectness of the factors employed inevaluating the probability functions, the curvesmay be used for estimating the demand loadsfor any particular number of fixtures of onegiven kind. However, the design load for allkinds of fixtures installed in one system shouldnot be the sum of the design loads computedseparately for each kind of fixture, even thoughthe individual curves may be correct. Simul­taneous operation of different killds of fixturesis a chance occurrence which would have to beevaluated by another probability function.Although such an evaluation is possible all thebasis of the assumed average time factors, theprocess is too complicated for convenient appli-

[ 12 ]

3.353.904.304.42

3.99

TABLE 3.-Relative demand weights of fixtures

N either the possible accuracy in estimatingthe demand to be expected in a plumbing sys­tem nor in estimating the capacity of pipesselected for supplying the estimated demandjustifies the assignment of weights in the scale1 to 10 closer than to the nearest integer.Accordingly, on the basis of average values, therelative weights of the three fixtures become 10for flush valves for water closets, 4 for bathtubs(total hot- and cold-water supply), and 5 forflush tanks.

It will be observed that these fixture weightshave the same significance as the fixture unitratings assigned to the different kinds of fix­tures in Recommended :Minimum Require­ments for Plumbing [1]. This unit in the weightscale is not a definite unit of flow but is simplya load factor to be applied through a relationbetween m and n such as developed in figures2 and 3. However, since the termjixture unitin this significance has become fairly well es­tablished by usage during the past 15 years, theterm will be retained here to designate loadweights of fixtures.

A further comparison and an illustration ofthe results of applying the average weights ofdifferent kinds of fixtures to a single basicprobability curve may be made by reducing thenumber of fixtures to fixture units (n timesweight) and replotting, as in figure 4. It willbe observed that the curves for flush tanks andbathtubs (total supply) lie very close together,indicating that the relative weights, 5 to 4, areapproximately correct throughout the rangeillustrated for the individual volumes and ratesallowed. It will also be observed that the twocurves both intersect the flush-valve curve be­

tween 193 gpm 880 fixture units and 210 gpm1,040 fixture units, indicating again that therelative weights 10:5:4 are approximately cor­rect in that range. For 300 flush tanks, 1,500

cation in thc field or office and, because thetime factors for faucet-supplied fixtures onwhich the probability of simultaneous operationdepends cannot be determined in general,would in the end lead to doubtful results. For

these reasons it is believed that reasonablysatisfactory results, whieh are much simpler toapply, may be obtained by weighting eachkind of fixture and applying the weighted sumof the total nmIl bel's of fixtures of all kinds to aload curve for flush valv{~s or a load curve forfhlsh tanks, according to which type of supplyis to be used.

4. DERIVATION OF FIXTURE WEIGHTS (FIXTURE

UNITS)

In I~ecommended 1'vIinimum Requirementsfor Plumbing [1], fixtures were weighted in thescale from 1 to 6, this scale being selected

. largely because, of the fixtures installed in thegreatest numbers, the smallest load (lavatory)was estimated to be about one-sixth of thelargest load (water closet). The scale chosenis purely arbitrary, and it is now suggestedthat a decimal scale, 1 to 10, would give a muchmore flexible system of estimating.

A comparison of the relative loading effectsof the three kinds of fixtures may be made bymeans of the curves in figure 3. Obviouslythere is no eommon exact ratio of the loadingeffects of any two different kinds of fixtures,each kind using water at different volumes, Q,volume rates, q, and different time intervals, T,over any eonsiderable range in number offixtures, n. For example, for a demand of.'50 gpm, the number of fixtures from whieh thisload would be oecurring not more than 1 percentof the time is approximately (fig. 3) as 5: 34: 33,respectively, for flush .valves for water closets,flush tanks for water closets, and bathtubsupply (total). The load-producing weightsare inversely as the number of fixtures pro­dueing a given load; hence, on the basis of10 for flush valves, the weights for a demandof 50 gpm are 10: 1.47: 1.52. Table 3 givesthe relative weights computed in this mannerfrom a comparison of the three curves fordemand loads at increments of 50 gpm from150 to 300 gpm, the range in which the pro­posed method of estimates will have its great­est application and usefulness.

[ 13 ]

Flush valvesFlush tanksBathtnhsDemand.

~---'------------.----------gpm

Number NnmherRelativeNUInberRelalof fixture~ Weightof fixturesweight

of fixtureswei~

n n n----- --------------------------150. __.

56101334.20167200

93101855.03238250. 133102385.59312300 ... __.. 176102926.03398--- ------~------

Average weighL_ 10---------- 5.21----------

IVP

ht

-

./

I. FLUSH VALVES 2. FLUSH TANKS

./,/3. BATH TUBS

.//'

,/

./

VI"

)/

I

,/'./

,/

/./V

//3./

/~

~

;;.-'

~~/'

~

/I~/{/

/'#

, /ij'///~

'I;500

400

300

200

100

oo 500 1000 1500

FIXTURE UNITS

2000 2500 3000

FIGURE 4.-Relation of demand to fixture units.

fixture units, referred separately to the flush­valve curve for an estimate, the demand esti­mate would be about 15 percent lower than thecorresponding estimate made directly from thefiush-tank curve. The corresponding error inthe estimate for 300 bathtubs made in the sameIwumer would be about 3}~percent. Theseerrorsare immaterial, for the only result, in case thedesign load was exceeded in service by thatamount, would be an increase in the time re­quired to refill the fixtures by 15 percent and3}~percent, respectively, or, in case the sametime is occupied in refilling, a reduction by thesepercentages in the volume of water used. Be­low the point of intersection, referring flushtanks and bathtubs separately to the flush­valve curve would result in overestimating thedemand by amounts varying from very smallpercentages for 880 to 1,040 fixture units to

about 94 and 23 percent, respectively, for 100fixture units. However, the error in an esti­mate made from curve 1 fol' the total demandload for flush valves for water closets and forbathtubs will be less than the error indicatedby an estimate made separately for the bath­tubs from the same eurve in all cases in whiehthe flush valves predominate, 011 the basis oftotal fixture units of the two kinds of fixtures.In cases where flush tanks for water closets arcused exelusively or predominantly in the sys­tem, a closer estimate could be made on thebasis of total weights, by using curve 2 and thetotal fixture units for all kinds of fixtures in­volved. Obviously the error made by usingcurve 2 for both flush tanks and bathtubs forany number of either up to 300 would be small.Also, the demand load relative to the numberof fixture units may be approximately repre-

[ 14]

500

/"/"

,./

./

./

/"

,v./. .//"./

./V./ .///'

-::?

./';/

IV/

V2V

V/

V//

fj I

100

200

400

~ 300a.<.9IoZ<t~Wo

oo 500 1000 1500

FIXTURE UNITS

2000 2500 3000

FIGUlU} ;).-E."/i1llale curve for design purposes.

sented in this range by a smooth curve drawnabov<, the two probability curves and mcrgedwith curve 1 as shown by the broken line infigure 4, thus giving estimates slightly in excessof the peak demands indicated by the separatec1l!'Yesfor flush tanks and bathtubs. The bro­ken line in figure 4 is reproduced in figure 5, to­gdher with curve 1. Thc curves in this figureare proposed for estimating design loads forwa ter-supply lines in general, curve 1 to beused when flush valves predominate in the sys­tem and curve 2 to be used when flush tankspredominate, the common c1l!'Ye above thebranch to be used for all weighted fixtures.

Of the fixtures commonly installed in thegreatest numbers, these three produce the majorpnrt of the peak demand. Lavutories are in­stalled in most systems in us grca t numbers, and

frequently in greater numbers, but obviouslyhave a much smaller load-producing weight, andare frequently ignored in estimating demandand sewage loads. Because, as pointed out indiscussing supply demands for bathtubs, it isimpossible to estimate the values of t and Treliably for faucet-supplied fixtures, and becauseof the relatively small effect on the totaldemand, it is suggested that satisfactoryfixture-unit ratings may be assigned to irregu­larly used faucet-supplied fixtures independ­ently of the probability function from aconsideration of the sizes of the supply outletsand the relative quantities of water used. Onthe basis of this reasoning, the relative weightssuggested for the kinds of fixtures most com­monly subjected to congested service are:10:5:4:2 for flush valves for water closets,

[ 15 ]

TABLE 4.-])i~trib/ltio1l of lIIo/"n1'ng calls in hotels

It will be observed that the time of risingin the two hotels is distributed in approximatelythe same manner, the greatest Humber of callsin any 15-minute perioo being for that follow­ing 7:30 11. m., and that slightly more than

two classes of supply should be added to obtainLlle esLimnLe for LoLn1 demand on any supplypipe common to both services. If the two typesof demand do not come at the same time of day,the greater demand of the two may be tflken asthe peak demand, or design estimate.

A bathroom group of fixtures can probably bemore closely weighted to the selected scale of 1to 10 by taking the three common bathroomfixtures-water closet, lavatory, and bathtub orshower-as a unit than by weighting eflchfixture separately. In hotels and apartmenthouses the bathrooms arc not all in use at onetime, and the loading weight of any fixture ofthe group-for example, a water closet-willbe much lower than for the same fixture incongested service, for which it has been assumedthat all fixtures will be in continuous use ouringthe period of congestion. One peak perioooccurs in hotels and apartments in the morningduring the rising hour and another in the after­noon during the preparations for dinner.Probably the best indication of the distributionof the use of the bathrooms in hotels and apart­men t buildings obtainable will be given byrecords of the activity during the hours whenthe peak use of the plumbing fixtures occurs.Table 4 is a summary of reconled morning(rising) calls from the switchboard for two largehotels in N ew York Ci ty for 7 consecu tive daysi]] one case and 9 days in the other. Hotel] isa large hotel in which the guests were principallytransient. Hotel 2 is a large apartment hotelin which the greater part of tl](' guests wereresident but which also received and accom­modated transients.

Till1f' of ('all

4810[14101 700[18o[ '1I::3[ :3821 2,8102,1)99 I jf) j"RO 1il) ·142 ;{41 3, Ofii

,ORO[ :J2211,4ROj :3011 OO,oj n31,o.HlO1

--

Be·

fore6::301\:4,\ '

5:~O --------

]74

22898

198 212113~m4AO[2311

Hotel L_u _Hotol2 __u _

Total cal1su_

Place

5. RELATIVE LOAD VVEIGHTS FOR DIFFERENT

CONDITIONS OF SER VICE

flush tanks for water closets, bath tubs. andlavntorics, rcspectively.

Up to this point the discussion has been con­fined to four kinds of fixtures under congestedconditions of service. There are several con­siderations that should be taken into account indetermining load ratings for other fixtures andfor other conditions of service, among whichthe following are especially important:

(1) Fixtures that are relatively few in numberand unlikely to be used when the predominantkinds are being used most frequently will addvery little to the demand or to the peak sewageload and hence may be ignored, except in regardto the branch supply lines and branch drainpipes of these fixtures. Slop or service sinks inoffice buildings, which are in use to any con­siderable extent only before or after office hours,add a negligible load to the peak loads of theday. Kitchen sinks and laundry trays indwellings and residential apartments may alsobe placed in this category.

(2) Fixtures so installed that they cannot ingeneral be subject to congested conditiolls ofservice in the same sense as fixtures installed inpublic comfort stations, general toilet rooms inoffice buildings, and other buildings in whicheach fixture is open and accessible for use ft t, alltimes, should be given a rating in accordancewith the possible extent or frequency of theiruse. Bathrooms in private dwellings, resi­dential apartments, and private bathrooms inhotels may be considered in this class, and canbe rated advantageously as a group.

(3) 'Vater services that demand a con tinuousflow, such as lawn sprinklers, air-conditioningequipment, gang showers in factories, andathletic dressing rooms, present no element ofchance in regard to overlapping and arc notsusceptible of a logical weighting in relation towater closets and other fixtures that use wflterat high rates for comparatively short periods oftime. Hence the demand for this type ofsupply should be considered separately and esti­mated separately. If the use of these continu­ous water supplies is such that they ov('rlflp therush period of the day for the weighted fixturesin the system, the separate estimates for the

[ 16]

TABLE 6.-Distribution of morning exits in apartmentsfor 2 days

TABLE 5.-Distribution of morning calls in apartmentsfor .9 days

one-half of all calls fall within the hour from7: 00 to 8: 00 a. m. This is significant, andit may be assumed that the time of rising ofguests leaving no morning call will be similarlydistributed in time.

Time of call

Period

Tota]Be- After

callg

fore6:306:457:007:157:307:458:00

6:30

8:00

-----------------

Igt daY ____n ______n327113631 137

2d daYnnn_nn_n848122931 249

3d daYn __________n629123932 248

4th daY ___nnnnn

439123831 1445th daY ______n _____

329122831 1416th daYnnnn __n_

547112821 141

7th daYnnnnn_n

8471331142 1538th daY_nnnnnn

4491141122 3509th daYn_n_n __n_

43101241031 148-- -------------Tota1 calls __n_n_

45287.\1062680261213411

tensive use of plumbing fixtures in any singleapartment or in any group of apartments inwhich the occupants rise at the same hour willlast for about an hour. The maximum number

of calls within any single hour ranges from 69to 75 percent for different days. The corre­sponding percentages for the hotels range from47 to 62. It therefore seems reasonable toassume that not more than 75 percent of thebathrooms in an apartment house or in anyhotel will be occupied and in use at one time.There may be exceptional apartment houses inwhich the occupants are all engaged in thesame occupation that would show a higher con­centration in rising time. However, it will bethe exception rather than the rule andshould be treated as an exception in estimatingwater-supply demands and sewage loads.

There is another consideration that is impor­tant in estimates of loads for bathrooms. In aleisurely use of a bathroom, the fixtures willordinarily be used one at a time. In cases ofhurry or congestion, two fixtures may be inuse at the same time in a bathroom, in the sensethat water is being drawn for or is being dis­charged from both at the same time. Henceit seems reasonable to assume that, on theaverage, not more than two-thirds of the totalnumber of fixtures in all occupied bathroomswill be in use at the same time. The totalnumber in use at one time on this basis, will notexceed two-thirds of 75 percent, or 50 percent,of the total number of fixtures in bathrooms inthe building. Accordingly, it is suggested thata bathroom group in a residence or apartment,or a private bathroom in a hotel be given one­half the total weight for the same fixtures inpublic or congested service in estimating supplydemands.

The term "bathroom group" is here definedas consisting of a water closet, a lavatory, anda bathtub with or without a shower head, or ofa water closet, a lavatory, and a shower stall.These suggested weights become (10+4+2)-+­2=8, and (5+4+2)-+-2=5.5, respectively, forinstallations using flush valves and flush tanks.Again, in accordance with the convention pre­viously used in selecting fixture weights, thenearest higher integral value will be assigned,giving for a bathroom group using a flush tankthe weight of 6.

47 I 70 I 41 I 31 I 26 I 27949 59 .54 23 21 262

96 I 129 1 95 1 54 I 47 1 541

Tota1

7:4518:00 18:1518:30 18:451 exitsto to to to to

8:00 8: 1.\ 8:30 8:45 9:00

Interva] of time

4.\ 7:00 7: 157:300

tototo007: 15

7:307:45--

-----I

418316

81527------

71233586

1,

7:

Period

1st daY n 1 12d daYnnnnn

Tota] exits. __.I 1

Table 5 gives the summarized record ofmorning calls for a large apartment house in·Washington, D.O., for 9 different days otherthan Sundays during a period of 22 days. Table6 gives the summarized record of morningexits from the same apartment house. Oom­paring tables 4 and 5, it will be observed thatthe number of morning calls in 15-minuteperiods in hotels and apartments fluctuates inabout the same manner, the principal differencebeing that the peak number of calls in any 15­minute period is about 10 to 15 percent of thetotal number higher for the apartment housethan for either the transient or apartmenthotel. Also comparing tables 5 and 6, it willbe observed that the peak of exits from theapartment house, indicating the beginning ofthe business or social activity of the day, occursapproximately 1 hour later than the peak ofthe rising activity. This indicates that ex-

[ 17]

~ See discussion of negligible fixtures on page 16.

TABLE 7.-Demand weights of plumbing fixtures

6. ApPLICA'rION OF LOAD CHART AND WEIGHT

TABLE

In estimating the demand load for a supplypipe in any building, the total number of eachkind of weighted fixtures or weighted groups offixtures will be multiplied by the weight of thatfixture or group (table 7) and the products addedto obtain the total number of fixture units.The demand load is given by the correspondingordinate of the appropriate curve in figure 5.For example, assume that an apartment houseor a hotel has 100 bathrooms with flush-tanksupply for all water closets and that any otherfixtures in the building are negligible 4 in rela­tion to peak demand. The total number offixture units will be 100X6, or 600. The ordi­nate on curve 2 of figure 5 corresponding to thisabscissa is 147 gpm. This estimate, of course,applies only to the main supply or service pipefor the building and does not include any con­tinuous demand, such as that for lawn sprinklersor air conditioning, etc. Again, assume thatan office building has 100 water closets withflush-valve supply, 25 stall urinals with flush­valve supply, and 100 lavatories, and that thedemand of other plumbing fixtures is negligiblein relation to these, but that the building hasan air-cooling system which demands a maxi­mum rate of 225 gpm. By referring to table

105

105321.5

4

34

3

866431.5

Wcir,htpcr fix·ture or

gronp infixtureunits

Type of supply

Flush valve (total) n_Flush valve (cold only)_Flush tank (total)_ . ..Flush tank (cold only) n

~3~wf~erc~~;:spon(linggroup above for total.2; for cold or hot

Occu·pancyFixture or group

Bathromn group un Private00_" 0000_'_0000 do. __ .._Do _nun_n. __h_ 00.<10 _

Do do. __u

00_ . 0000_._0000 do __ u

Bathroom r,roup with _.._do. _00

separate shower.

Water closet uunn_ Puhlic __ Flush valve nun_Do 0000_000000 n_do _00 Flush tank_.nnn _

Pedestal urinaLnu __n do Flush valvcn_nnn _Stall or wall urinaL_on n_do. _00 n do _00000000_0000_00

DO_._nnn_n .._n do_on Flush tank __nnnnnnLavatorYnnnnn_nn do __00 Total... 00_000000 __ 00_

Do. _"n_nnn'_h_ 00_<10 Hot or cold un_nnBathtuhsnnnnnnn __ dO_on TotaL_unnhhn n

DO.__unnnnnn do llotorcold __. nh_Shower head .. h. __ do_ TotaL_n nnnnnn

00 0000 __ '00_00_00 do _. Hot or cold unn __. __

In suggesting this rating for a bathroomgroup, it is l'eeognized that the condition ofservice for which the rating is made does notcompletely block out of use one-half of eachkind of fixture, as would be required for theresult obtained to be exact. What actuallyoccurs in the installations to which the proposedrating would apply is that one-fourth or moreof the fixtures of each kind are not in serviceduring the time the other three-fourths or lessare in actual use, and that the ratio of tiT foreach kind of fixture is greatly decreased belowthe ratio applying to congested conditions ofservice. For example, if each of the water­closet flush valves is operated twice during thehour in which it was assumed that two-thirdsof the bathrooms were in use, t/T=9/1800=1/200; if used four times, t/T=9/900=1/100, ascompared to t/T=9/300=3/100 for congestedservice. A similar relation will exist betweenthe ratios of tiT for flush tanks and bathtubsunder the conditions described and undercongested conditions. It may be noted in thisconnection that assigning a bathroom groupone-half the rating that is given the same fixturesin congested service would give an estimate ofthe demand greater than one-half of that forcongested service. For example, 10 flushvalves for water closets, 10 bathtubs, and 10lavatories in congested service total 160 fixtureunits, which by curve 1 in figure 5 gives anestimated demand of 81 gpm. The proposedrating for 10 bathroom groups with flush valves,80 fixture units, gives an estimated demand of64 gpm.

Table 7 gives the fixture weights suggested inaccordance with the use to which the fixturesare subjected and the manner in which theyare installed. The term" public" refers to fix­tures which are individually open for use at alltimes when the building is open, as in publictoilets or general toilets in office buildings."Private" refers to fixtures installed in groupsin such a manner that the entire group may beand generally is confined to the use of one per­son at a time, as in residences or private bathsof hotels. "Total" refers to hot and coldsupply combined. "Hot or cold" refers to hotor cold water supply only.

[ 18 ]

560

5 it is found that the total number of fixtureunits is 100XIJ+25X5+100X2, or 1,325.The corresponding estimated demand load is 247gpm, to which 225 gpm must be added, giving472 gpm for the total estimated demand on theservice pipe of the building. The method ofestimating for branch risers and other dis­tributing pipes is similar, the estimate beingbased on the' number of fixtures or groupssupplied through the branch, and the weightsbeing selected a.c(~ording to whether the branchsupply pipe carried the total supply, the coldwater only, or the hot water only.

X. ADEQUACY OF THE IvIETHOD

The proof of the adequacy of the proposedmethod of estimating the demand loads to beexpected in building-water-supply systems will,in the end, depend on its success in actual trialover a period of years. Fortunately, in thiscase we have a means of comparison with asimilar method that has been given a trial.

Table 8 is an abridgement of a table from theconstruction manual [5] used by the mechanicalengineering section of one of the Federal de­partments, based on the method of estimate bymeans of the probability function as originallyproposed [1] in 1923. This table has been usedfor a number of years by that office with reason­ably successful results. The only complaintagainst the results of using this table and methodthat has come to the author's attention is thatit tends to give larger estimates than havebeen found necessary for satisfactory service.The principal reason for this tendency to over­size supply pipes does not lie in any inherentfault in the probability function, but in theapplication of the method by means of a tablewhich does not provide for the probability,or rather the improbability, of overlapping be­tween or among two or more groups of differentkinds.

TABLE S.-Demand estimates in gallons per minutefrom li'ederal table

Number of fixtures

The inherent fault of applying the proba­bility function by means of a table is illus­trated in the comparisons which follow. Table9 gives estimated demand loads for flush valvesfor water closets, flush valves for urinals, flushtanks, and lavatory faucets, taken from thecurves of figure 4 over the range in number offixtures included in table 8. The values forflush valves for urinals were taken from thecurve for flush valves for water closets on thebasis of five fixture units per fixture, and thevalues for lavatories from the curve for bath­tubs on the basis of two fixture units per fixture(table 7).

TABLE g.-Demand estimates in gallons per minute frO/iifigure 4

)<umber of fixturesKind of fixture

-;~ 12 162~~;;f100 150~-;;~~

- - - - - - - - - - ---Water·closet flush tanks. 00 __

81218243039 56 66 96117167216310Water·closet flush va]ves ......

3647161 7082981251401182205269327435Urinal flush va]ves .... n ,, ____

27364754 61 71 89 98125140175205269Lavatoriesu u_hhnnhn __

8 121620 23 29 39 43 58 fi5 84 102137

There are obvious irregularities in the esti­mates given by table 8, especially in the esti­mates for flush tanks and lavatory faucets.This may be seen by plotting these values orby comparing them by steps as is done intable 10.TABLE 1O.-Demand estimates in gallons per minute per

fixture by table 8

Xumber of fixturesKind of fixture First 50

SecondThird 50

FourthLast IC50 50---------------

F]ush tanks .. _00 ••

1.160.740.901.50O.

Lavatories ___.. _.0.72.48.300.20

This makes the irregularities in table 7 obvi­ous. To be consistent, the increment in theestimate for a given increment in number offixtures should gradually approach a constantminimum as the total number increases.

Table 11 shows the regular decrease in de­mand per fixture as given in table 9.TABLE ll.-Demand estimates in ga.llons per minute per

fixture by table 9

Xumher of fixture~

Kind of fixture 1--------- ..------------------·­first 50 I second 50 I third 50 I fourth 50 I last ]00

Kind of fixture I---J-'

2 4 8 It" 24140 50 80 100150200300------- ~ - -- - - --- -.- - --- ---- -- .~-~

Water·closet lIush tanks. nn 8 12 IG 20 30 38 50 58 80 95 11021.' 290

Water·closot flash v'llves ----1361661 72)" 'ool,~,~ '00 ""I~.~,~,Urinaillush valves. __ ._n __• 182.330.3642 50/64 72 90 10012511818~La'·atones. nnn •• __ n 610]51820 2.5 30 36151 601 75 85110, I I

F]ush tanks ... __..Lavatories __nu _

1.320_86

1.020.54

1.000.38

0.98.36

0.94.35

[ 19]

The regularly d<,creasing allowances perfixture in this table arc propel' and permissible.However, although the demand pel' fixture de­creases with increasing number of fixtures,tending toward the average demands for entiresystems, the estimate per fixture for satis­factory service of the entire group of fixturescannot fall to that averagl', approximately 0.8gpm per water closet and 0.16 gpm per lavatory,if the supply pipes aTe to be capable of supplyingthe peak demands.

There arl' two particular requirements thatany practical method of estimating demandloads should meet. First, the method shouldgive estimates greater than thl' averag<' demandfor all fixtures in the system during the periodsof heaviest use; otherwise complete interrup­tion of service ",ill occur. Second, the methodshould give satisfactory and economical esti­mates for groups of fixtures all of the same kindand for groups composed of any number ofdifferent kinds of fixtures. Thl' inherent de­fects of a tabll' in the form of tables 8 or 9 inml'eting the second requirement will becomeapparent from the comparisons given in table 12.

TABLE 12.-Comparison of demand estimates for plumb­ing systems

taken does not specifically state that the SUIll

of the separate estimates for each kind offixture is to be taken as the total estimate, itdoes not provide any other means of obtainingthe total, and presuma bly this is the methodthat has been used in applying the table. Ifthis is true, it accounts to a large extent for theoverestimates complained of and previouslyreferred to. Furthermore, no provision ismade in the manual for the difference in demandfor the same fixtures in public (congested)service and in private (noncongested) service,an omission that would still further increasethe overestimates for the latter.

In addition to the fact that the method ofweighting fixtures as to their demands (table 7)and applying the weighted total to probablense curves (fig. 5) will at least paTtially correctthe errors inherent in the use of tables asillustrated, once the weighted method is under­stood thoroughly, it will be simpler to use.Tables in an abbreviated form, as illustratedby tables 8 and 9, require interpolation forvalues not appearing in the tahle, whereasinterpolation is automatically taken care of inusing the curve.

XI. ESTIMATES OF SEWAGE LOADSEstimates by

" ",,,ter closet." Urinal.

TotaL 00 1~1--8G-'_____;;;_~50 flush v"lves CW. C.) h h_h 500 I-c----h 140 14324 flush v"I'I"OSCUr.) h hh 120 uh 71 50SOlavatories nn un_hdun __ 100 u 43 36

200 flllsb valves C"-. 0.) . 00001 2.000 1_000_000 327 29.)100 IIlIsh v~l\"es (Ur.) h h_ .>00 __00 __ 00 140 lex.!

200 ]"vatones h Oo hh_I~::.:.::.:.:.:.: __ 10:" 8_"TotaL h h hh_ 2.900 I 432 569 480

Table 7 and Table I Tablefigure 5 9 8

______________ ._I_fi_I~_~_\\_~e_l_g_p_m__ g_p_m_l_g_p_m_

It has already been pointed out on pages4 and 5 that the probability function has onlya limited application to the drainage systemsof buildings. Insofar as the probability func­tion is applicable to the estimation of peaksewage loads to be provided for, the same fUllc­tion as applied to the peak water demand (fig.5) will apply also, with certain modifications,to the drainage system. In the main, tIll' sametable of fixture weights (fixture units) will alsoapply to both the supply and drainage systems.The principal exception is tll(' water closet,which for the drainage system will in generalhave the same weight for a given congestion ofservice whether the water-supply control is b."flush valve or flush tank. In certain othercases, it may become advisnblr to assign dif­ferent weights to the same fixture, depl'ndingon the sizes of the outlet from thl' fixture andof thl' fixture drain. Because the details of themodifications indicated are dosl'ly relatrd to tlH'capacities of drains, the further development.

723018

22\1

704720

2M171

120 100_0000_

40 00 00 00 00

24 00_00

720TotaL _00 00_ 00 __

Number and kind of fixture,

12 flusb valves (W. C.) " __00_0000 __ 00

8 flush v"lves (Ur.) ,, 0000 _

12 lavatories n _

A comparison of these estimates shows clearlythe tendency to overestimate when the separateestimates for the different kinds of fixtures areadded directl~v to obtain the estimate for totaldemand for all fixtures. Although the manualfrom which the estimate values in table 8 were

r 20]

of the method of estimating sewage load willbe treated in the paper dealing with capacitiesof building drains. Also, before the methodcan be successfully applied in practice, theabbreviated table of fixtme weights for demandloads will have to be expandpd and modifipd.Thesp modifications will also be developpd inthe lakr paper.

XII. DISCUSSION

The principal purpose of this paper is topresent the fundamental principles and data onwhich the proposed method of estimatingloads to be provided for in plumbing systemsis bascd. It, has been pointed out that thedetails of application of any method in practicemust be guided to a large extent by engineeringjudgment in order that it may lead to satis­factory results. The choice of values employedin evahmting the probability function and inconverting estimates in number of fixturesflowing to estimates in gallons per minute rep­resents the author's judgment.

It, has also been pointed out that it is impos­sible to determine or to estimate closely eitherthe maximum demand load or the maximulllsewage loael that will occur in service. It isonly possible to estimate the loads having acertain probability of being exceeded in a givensystem. In this respect, it is believed thattable 7 and figure 5 will enable the engineer toobtai n t,hat, estimate as closel~Tas the vnr~Ting

o

conditions encountered in plumbing systemspermit, and that the results of applying thedata presented in that form will in general leadto the selection of adequate sizes of supplypipes and as economical sizes as are consistentwith safety and satisfactory operation, pro­vided the proper judgment is exercised inestima ting the capacity of the pipes under theparticular conditions of installation.

XIII. REFERENCES

[1] Recommendcd Minimum Requirements for Plumb­ing, Departmcnt of Commercc Building andHousing Series, BH13, 1924. Revised by addi­tion of a supplement, 1932.

[2] Probability and Its Engineering Uses, by ThorntonC. Fry, (D. Van Nostrand and Co., Inc., 1928).

[3] The Hydraulics of Water Closet Bowl.~ and FlushingDevices, by Thomas R. Camp. associate professorof Sanitary Engineering, Massachusetts Instituteof Technology. Published in mimeographedform in 1936 by the Massachusetts State Asso­ciatiou of Nlaster Plumbers.

[4J G. E. Golden, Research Associate at the NationalBureau of Standards for the Plumbing Manu­facturers Research Associateship. (Data un­published.)

[5] Manual of Procedure for the Mechanical Section,Public Works Branch, Procuremcnt Division,Treasury Department (Sept. 1935).

[6] Interior Water Supply Piping for ResidcntialBuildings, Bulletin of the University of Wiscon­sin, Engineering Expcriment Station Series, No.77.

WASHINGTON, August 30, 1940.

[21 ]

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